LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

Class 


Machine  Design 


A  Manual  of 

PRACTICAL     INSTRUCTION     IN     THE    ART    OF    CREATING    MACHINERY    FOR 
SPECIFIC    PURPOSES,   INCLUDING  MANY  WORKING  HINTS  ESSEN- 
TIAL  TO    EFFICIENCY    IN    THE    OPERATION    AND    CARE 
OF    MACHINES,    AND    INCREASE    OF    OUTPUT 


By  CHARLES  L.   GRIFFIN,  S.B. 

American  Society  of  Mechanical  Engineers.     Mechanical  Engineer  with 

the  Semet-Solvay  Company.     Formerly  Professor  cf  Machine 

Design,  Pennsylvania  State  College. 


ILLUSTRATED 


CHICAGO 

AMERICAN    SCHOOL  OF   CORRESPONDENCE 
1908 


COPYRIGHT  1907  BY 
AMERICAN  SCHOOL  OF  CORRESPONDENCE; 


Entered  at  Stationers'  Hall,  London 
All  Rights  Reserved 


Foreword 


recent  years,  such  marvelous  advances  have  been 
made  in  the  engineering  and  scientific  fields,  and 

O  o 

so  rapid  has  been  the  evolution  of  mechanical  and 
constructive  processes  and  methods,  that  a  distinct 
need  has  been  created  for  a  series  of  practical 
,  of  convenient  size  and  low  cost,  embodying  the 
accumulated  results  of  experience  and  the  most  approved  modern 
practice  along  a  great  variety  of  lines.  To  fill  this  acknowledged 
need,  is  the  special  purpose  of  the  series  of  handbooks  to  which 
this  volume  belongs. 

C,  In  the  preparation  of  this  series,  it  has  been  the  aim  of  the  pub- 
lishers to  lay  special  stress  on  the  practical  side  of  each  subject, 
as  distinguished  from  mere  theoretical  or  academic  discussion. 
Each  volume  is  written  by  a  well-known  expert  of  acknowledged 
authority  in  his  special  line,  and  is  based  on  a  most  careful  study 
of  practical  needs  and  up-to-date  methods  as  developed  under  the 
conditions  of  actual  practice  in  the  field,  the  shop,  the  mill,  the 
power  house,  the  drafting  room,  the  engine  room,  etc. 

C,  These  volumes  are  especially  adapted  for  purposes  of  self- 
instruction  and  home  study.  The  utmost  care  has  been  used  to 
bring  the  treatment  of  each  subject  within  the  range  of  the  com- 


179729 


mon  understanding,  so  that  the  work  will  appeal  not  only  to  the 
technically  trained  expert,  but  also  to  the  beginner  and  the  self- 
taught  practical  man  who  wishes  to  keep  abreast  of  modern 
progress.  The  language  is  simple  and  clear;  heavy  technical  terms 
and  the  formulae  of  the  higher  mathematics  have  been  avoided, 
yet  without  sacrificing  any  of  the  requirements  of  practical 
instruction;  the  arrangement  of  matter  is  such  as  to  carry  the 
reader  along  by  easy  steps  to  complete  mastery  of  each  subject ; 
frequent  examples  for  practice  are  given,  to  enable  the  reader  to 
test  his  knowledge  and  make  it  a  permanent  possession;  and  the 
illustrations  are  selected  with  the  greatest  care  to  supplement  and 
make  clear  the  references  in  the  text. 

C,  The  method  adopted  in  the  preparation  of  these  volumes  is  that 
wrhich  the  American  School  of  Correspondence  has  developed  and 
employed  so  successfully  for  many  years.  It  is  not  an  experiment, 
but  has  stood  the  severest  of  all  tests — that  of  practical  use — which 
has  demonstrated  it  to  be  the  best  method  yet  devised  for  the 
education  of  the  busy  working  man. 

C.  For  purposes  of  ready  reference  and  timely  information  when 
needed,  it  is  believed  that  this  series  of  handbooks  will  be  found  to 
meet  every  requirement. 


Table    of    Contents 


PRINCIPLES  AND  METHOD Page   3 

Object  of  Machine  Design — Mechanical  Thought,  Develop-mesrt, — and 
Specification — Importance  of  Details — Relation  of  Design  to  Problems 
it  Seeks  to  Solve — Theory  and  Production — Invention — Use  of  Hand- 
books and  Empirical  Data — Calculations,  Notes,  and  Records — 
Sketches — Method  of  Design — Analysis  of  Conditions  and  Forces — 
Theoretical  Design — Practical  Modification — Delineation  and  Specifi- 
cation— Constructive  Mechanics — Forces,  Moments,  and  Beams — Ten- 
sion, Compression,  and  Torsion — Friction  and  Lubrication — Working 
Stresses  and  Strains. 


APPLICATION  TO  A  PRACTICAL  CASE Page  23 

Design  of  an  Elevator  Wire-Rope  Drive — Preliminary  Sketch — Rope 
and  Drum — Driving  Gears — Pulleys — Torsional  Moment  around  Shaft 
Axes — Calculation  of  Width  of  Belt — Length  of  Bearings — Height  of 
Centers — Data  on  Sketch — Sizes  of  Shafts — Preliminary  Layoui — 
Pulleys — Gears — Brackets  and  Caps — Drum  and  Brake — Base — Brake- 
Strap  Bracket — Foot-Lever — Gear  Guard — Brake-Relief  Spring — The 
General  or  Assembled  Drawing- — "Reversed"  Machine  Design. 


CLASSIFICATION  OF  MACHINERY        .  .       .       .       .       .    Page  62 

Machine  Tools  (Lathe,  Planer,  Milling  Machine,  etc.) — Motive-Power 
Machinery  (Steam  Engine,  Air-Compressor,  Steam  Pump,  etc.)  — 
Structural  Machinery  (Cranes,  Elevators,  Locomotives,  Cars,  Cable- 
W^ays,  etc.) — Mill  and  Plant  Machinery  (Rolling  Mills,  Mining  Ma- 
chinery, Crushers,  Stamps,  Drills,  etc.) — Use  of  Cast  and  Wrought 
Iron  and  Steel — Suggestions  on  Original  Design. 


DESIGN  OF  COMPONENT  PARTS  OF  MACHINERY      .       .       .       .    Page  75 

Belts — Strength  of  Leather  Belting — Horse-Power  Transmitted  by 
Belts — Speed  of  Belting — Material  of  Belting — Initial  Tension  in  Belt 
— Pulleys  (Rim,  Arms,  Hub) — Split  Pulleys — Special  Forms  of  Pulleys 
— Shafts — Simple  and  Combined  Stresses  of  Torsion,  Bending,  Com- 
pression— Deflection — Centrifugal  Whirling — Horse-Power  of  Shaft- 
ing— Bessemer,  Open-Hearth,  and  Nickel  Steel — Spur  Gears — Involute 
and  Cycloidal  Curves — Pitch  Circles — Mortise  Teeth — Shrouding — 
Hook-Tooth  Gear — Stub  Tooth — Web  Gear — Bevel  Gears — Worm  and 
Worm  Gear — Friction  Clutches — Couplings  (Flange,  Clamp,  Claw)  — 
Bolts,  Studs,  Nuts,  and  Screws — Keys,  Pins,  and  Cotters — Spline — 
Bearings,  Brackets,  and  Stands. 


INDEX Page  183 


MACHINE  DESIGN, 


PART  I. 

Definition.  Machine  Design  is  the  art  of  mechanical  thought 
development,  and  specification. 

It  is  an  art,  in  that  its  routine  processes  can  be  analyzed  and 
systematically  applied.  Proficiency  in  the  art  positively  cannot 
be  attained  by  any  "  short  cut "  method.  There  is  nothing  of  a 
spectacular  nature  in  the  methods  of  Machine  Design.  Large 
results  cannot  be  accomplished  at  a  single  bound,  and  success  is 
possible  only  by  a  patient,  step-by-step  advance  in  accordance 
with  well-established  principles. 

"  Mechanical  thought "  means  the  thinking  of  things  strictly 
from  their  mechanical  side;  a  study  of  their  mechanical  theory, 
structure,  production,  and  use;  a  consideration  of  their  mechanical 
fitness  as  parts  of  a  machine. 

"Mechanical  development"  signifies  the  taking  of  an  id^a  in 
the  rough,  in  the  crude  form,  for  example,  in  which  it  comes  from 
the  inventor,  working  it  out  in  detail,  and  refining  and  fixing  it  in 
shape  by  the  designing  process.  Ideas  in  this  way  may  become 
commercially  practicable  designs. 

"  Mechanical  specification  "  implies  the  detailed  description 
of  designs,  in  such  exact  form  that  the  shop  workmen  are  enabled 
to  construct  completely  and  put  in  operation  the  machines  repre- 
sented in  the  designs. 

The  object  of  Machine  Design  is  the  creation  of  machinery 
for  specific  purposes.  Every  department  of  a  manufacturing 
plant  is  a  controlling  factor  in  the  design  and  production  of  the 
machines  built  there.  A  successful  design  cannot  be  out  of 

o 

harmony  with  the  organized  methods  of  production.  Hence  in 
the  high  development  of  the  art  of  Machine  Design  is  involved  a 
knowledge  of  the  operations  in  all  the  departments  of  a  manu- 
facturing plant.  The  student  is  therefore  urged  not  only  to 
familiarize  himself  with  the  direct  production  of  machinery,  but  to 
study  the  relation  thereto  of  the  allied  commercial  departments- 


MACHINE  DESIGN 


He  should  get  into  the  spirit  of  business  at  the  start,  get  into  the 
shop  atmosphere,  execute  his  work  just  as  though  the  resulting 
design  were  to  be  built  and  sold  in  competition.  He  should  visit 
shops,  work  in  them  if  possible,  and  observe  details  of  design  and 
methods  of  finishing  machine  parts.  In  this  way  he  will  begin 
to  store  up  bits  of  information,  practical  and  commercial,  which 
will  have  valuable  bearing  on  his  engineering  study. 

The  labor  involved  in  the  design  of  a  complicated  automatic 
machine  is  evidenced  by  the  designer's  wonderful  familiarity  with 
its  every  detail  as  he  stands  before  the  completed  machine  in 
operation  and  explains  its  movements  to  an  observer.  The  intri- 
cate mass  of  levers,  shafts,  pulleys,  gears,  cams,  clutches,  etc.,  etc., 
packed  into  a  small  space,  and  confusing  even  to  a  mechanical 
mind,  seems  like  a  printed  book  to  the  designer  of  them. 

This  is  so  because  it  is  a  familiar  journey  for  the  designer's 
mind  to  run  over  a  path  which  it  has  already  traversed  so  many 
times  that  he  can  see  every  inch  of  it  with  his  eyes  shut.  Every 
detail  of  that  machine  has  been  picked  from  a  score  or  more  of 
possible  ideas.  One  by  one,  ideas  have  been  worked  out,  laid 
aside,  and  others  taken  up.  Little  by  little,  the  special  fitness  of 
certain  devices  has  become  established,  but  only  by  patient,  care- 
ful consideration  of  others,  which  at  first  seemed  equally  good. 

Every  line,  and  corner,  and  surface  of  each  piece,  however 
small  that  piece  may  be,  has  been  through  the  refining  process  of 
theoretical,  practical,  and  commercial  design.  Every  piece  has 
been  followed  in  the  mind's  eye  of  its  designer  from  the  crude 
material  of  which  it  is  made,  through  the  various  processes  of  fin- 
ashing,  to  its  final  location  in  the  completed  machine;  thus  its 
bodily  existence  there  is  but  the  realization  of  an  old  and  familiar 
picture. 

What  wonder  that  the  machine  seems  simple  to  the  designer 
of  it!  As  he  looks  back  to  the  multitude  of  ideas  invented, 
worked  out,  considered  and  discarded,  the  machine  in  its  final 
form  is  but  a  trifle.  It  merely  represents  a  survival  of  the  fittest. 

No  successful  machine,  however  simple,  was  ever  designed 
that  did  not  go  through  this  slow  process  of  evolution.  No 
machine  ever  just  simply  happened  by  accident  to  do  the  work 
for  which  it  is  valued.  No  other  principle  upon  which  the  sue- 


MACHINE  DESIGN 


cessful  design  of  machinery  depends  is  so  important  as  this  careful, 
patient  consideration  of  detail.  A  machine  is  seldom  unsuccessful 
because  some  main  point  of  construction  is  wrong.  Thejprincipal 
features  of  a  machine  are  usually  the  easiest  to  determine.  It  is 
a  failure  because  some  little  detail  was  overlooked,  or  hastily  con- 
sidered, or  allowed  to  be  neglected,  because  of  the  irksome  labor 
necessary  to  work  it  out  properly. 

There  is  no  task  so  tedious,  for  example,  as  the  devising  of 
the  method  of  lubricating  the  parts  of  a  complicated  machine. 
Yet  there  is  no  point  of  design  so  vital  to  its  life  and  operation  as 
an  absolute  assurance  of  an  adequate  supply  of  oil  for  the  moving 
parts  at  all  times  and  under  all  circumstances.  Suitable  means 
often  cannot  be  found,  after  the  parts  are  together,  hence  the 
machine  goes  into  service  on  a  risky  basis,  with  the  result,  per- 
haps, of  early  failure,  due  to  "running  dry."  Good  designers 
will  not  permit  a  design  to  leave  their  hands  which  does  not  pro- 
vide practically  automatic  oiling,  or  at  least  such  means  of  lubri- 
cation that  the  operator  can  offer  no  excuse  for  neglecting  to  oil 
his  machine.  This  is  but  a  single  illustration  of  many  which 
might  be  presented  to  impress  the  definite  and  detail  character 
necessary  in  work  in  Machine  Design. 

Relation.  The  relation  which  Machine  Design  should  cor- 
rectly bear  to  the  problems  that  it  seeks  to  solve,  is  twofold;  and 
there  are,  likewise,  two  points  of  view  corresponding  to  this  two- 
fold relation,  from  which  a  study  of  the  subject  should  be  traced. 
Neither  of  these  can  be  discarded  and  an  efficient  mastery  of  the 
art  attained.  These  points  are — 
I.  Theory. 

II.    Production. 

I.  Theory.  From  this  point  of  view,  Machine  Design  is 
merely  a  skeleton  or  framework  process,  resulting  in  a  repre- 
sentation of  ideas  of  pure  motion,  fundamental  shape,  and  ideal 
proportion.  It  implies  a  working  knowledge  of  physical  and 
mathematical  laws.  It  is  a  strictly  scientific  solution  of  the 
problem  at  hand,  and  may  be  based  purely  on  theory  which  has 
been  reasoned  out  by  calculation  or  deduced  from  experiment. 
This  is  the  only  sure  foundation  for  intelligent  design  of  any  sort. 

But  it  is  not  enough  to  view  the  subject  from  the  standpoint 


8  MACHINE  DESIGK 

of  theory  alone.  If  we  stopped  here  we  should  have  nothing  but 
mechanisms,  mere  laboratory  machines,  simply  structures  of 
ingenuity  and  examples  of  fine  mechanical  skill.  A  machine  may 
be  correct  in  the  theory  of  its  motions ;  it  may  be  correct  in  the 
theoretical  proportions  of  its  parts;  it  may  even  be  correct  in  its 
operation  for  the  time  being;  and  yet  its  complication,  its  mis- 
directed and  wasteful  effort,  its  lack  of  adjustment,  its  expensive 
and  irregular  construction,  its  lack  of  compactness,  its  difficulty 
of  ready  repair,  its  inability  to  hold  its  own  in  competition — any 
of  these  may  throw  the  balance  to  the  side  of  failure.  Such  a 
machine,  commercially  considered,  is  of  little  value.  No  shop 
will  build  it,  no  machinery  house  will  sell  it,  nobody  will  buy  it 
if  it  is  put  on  the  market. 

Thus  we  see  that,  aside  from  the  theoretical  correctness  oi 
principle,  the  design  of  a  machine  must  satisfy  certain  other 
exacting  requirements  of  a  distinctly  business  nature. 

II.  Production.  From  this  point  of  view,  Machine  Design 
is  the  practical,  marketable  development  of  mechanical  ideas, 
Viewed  thus,  the  theoretical,  skeleton  design  must  be  so  clothed 
and  shaped  that  its  production  may  be  cheap,  involving  simple 
and  efficient  processes  of  manufacture.  It  must  be  judged  by  the 
latest  shop  methods  for  exact  and  maximum  output.  It  must 
possess  all  the  good  points  of  its  competitor,  and,  withal,  some 
novel  and  valuable  ones  of  its  own.  In  these  days  of  keen  com- 
petition it  is  only  by  carefully  studied,  well-directed  effort  toward 
rapid,  efficient,  and,  therefore,  cheap  production  that  any  machine 
can  be  brought  to  a  commercial  basis,  no  matter  what  its  other 
merits  may  be.  All  this  must  be  thought  of  and  planned  for  in 
the  design,  and  the  final  shapes  arrived  at  are  quite  as  much  a 
result  of  this  second  point  of  view  as  of  the  first. 

As  a  good  illustration  of  this,  may  be  cited  the  effect  of  the 
present  somewhat  remarkable  development  of  the  so-called  "high 
speed  "  steels.  The  speeds  and  feeds  possible  with  tools  made  of 
these  steels  are  such  that  the  driving  power,  gearing,  and  feed 
mechanism  of  the  ordinary  lathe  are  wholly  inadequate  to  the 
demands  made  upon  tnem  when  working  the  tool  to  its  limit 
This  means  that  the  basis  of  design  as  used  for  the  ordinary  tool 
steel  will, not  do,  if 'the  machine  is  expected  to  stand  up  to  the 


MACHINE  DESIGN" 


cuts  possible  with  the  new  steels.  Hence,  while  the  old  designs 
were  right  for  the  old  standard,  a  new  one  has  been  set,  and  a 
thorough  revision  on  a  high-speed  basis  is.  imminenV-else  the 
market  for  them  as  machines  of  maximum  output  will  be  lost. 

From  these  definitions  it  is  evident  that  the  designer  must 
not  only  use  all  the  theory  at  his  command,  but  must  continually 
inform  himself  on  all  processes  and  conditions'  of  manufacture, 
and  keep  an  eye  on  the  tenderly  of  the  sales  markets,  both 
of  raw  material  and  the  finished  machinery  product.  This  is 
what  in  the  broadest  sense  is  nieant  by  the  term  "  Mechanical 
Thought,"  thought  which  is  directed  and  controlled,  not  only  by 
theoretical  principle  but  by  closely  observed  practice.  From  the 
feeblest  pretenders  of  design  to  .those  engineers  who  consummate 
the  boldest  feats  and  control  the  largest  enterprises,  the  process 
which  produces  results  is  always  the  same.  Although  experience 
is  necessary  for  the  best  mechanical  judgment,  yet  the  student- 
must  at  least  begin  to  cultivate  good  mechanical  sense  very  early 
in  his  study  of  design. 

Invention.  Invention  is  closely  related  to  Machine  Design, 
but  is  not  design  itself.  Whatever  is  invented  has  yet  to  be 
designed.  An  invention  is  of  little  value  until  it  has  been  refined 
by  the  process  of  design. 

Original  design  is  of  an  inventive  nature,  but  is  not  strictly 
invention.  Invention  is  usually  considered  as  the  result  of  genius, 
and  is  announced  in  a  flash  of  brilliancy.  We  see  only  the  flash, 
but  behind  the  flash  is  a  long  course  of  the  most  concentrated 
brain  effort.  Inventions  are  not  spontaneous,  are  not  thrown  off 
like  sparks  from  the  blacksmith's  anvil,  but  are  the  result  of  hard 
and  applied  thinking.  This  is  worth  noting  carefully,  for  the 
same  effort  which  produces  original  design  may  develop  a  valuable 
invention.  But  there  is  little  possibility  of  inventing  anything 
except  through  exhaustive  analysis  and  a  clear  interpretation  of 
such  analysis. 

Handbooks  and  Empirical  Data.  The  subject  matter  in 
these  is  often  contradictory  in  its  nature,  but  valuable  nevertheless, 
Empirical  data  are  data  for  certain  fixed  conditions  and  are  nor 
general.  Hence,  when  handbook  data  are  applied  to  some  specific 
case  of  design,  while  the  information  should  be  used  in  the  freest 


MACHINE  DESIGN 


manner,  yet  it  must  not  be  forgotten  that  the  case  at  hand  is  prob- 
ably different,  in  some  degree,  from  that  upon  which  the  data  were 
based,  and  unlike  any  other  case  which  ever  existed  or  will  ever 
again  exist.  Therefore  the" data  should  be  applied  with  the  greatest 
discretion,  and  when  so  applied  will  contribute  to  the  success  of 
the  design  at  least  as  a  check,  if  not  as  a  positive  factor. 

The  student  should  at  the  outset  purchase  one  good  handbook, 
and  acquire  the  habit  of  consulting  it  on  all  occasions,  checking 
and  comparing  his  own  calculations  and  designs  therefrom.  Care 
must  be  taken  not  to  become  tied  to  a  handbook  to  such  an  extent 
that  one's  own  results  are  wholly  subordinated  to  it.  Independence 
in  design  must  be  cultivated,  and  the  student  should  not  sacrifice 
his  calculated  results  until  chey  can  be  shown  to  be  false  or  based 
on  false  assumption.  Originality  and  confidence  in  design  will  be 
the  result  if  this  course  be  honestly  pursued. 

Calculations,  Notes,  and  Records.  Accurate  calculations  are 
the  basis  of  correct  proportions  of  machine  parts.  There  is  a  right 
way  to  make  calculations  and  a  wrong  way,  and  the  student  will 
usually  take  the  wrong  way  unless  he  is  cautioned  at  the  start. 

The  wrong  way  of  making  calculations  is  the  loose  and  shift- 
less  fashion  of  scratching  upon  a  scrap  of  detached  paper  marks 
and  figures,  arranged  in  haphazard  form,  and  disconnected  and 
incomplete.  These  calculations  are  in  a  few  moments'  time  totally 
meaningless,  even  to  the  author  of  them  himself,  and  are  so  easily 
lost  or  mislaid  that  when  wanted  they  usually  cannot  be  found. 

Engineering  calculations  should  always  be  made  systemati- 
cally, neatly,  and  in  perfectly  legible  form,  in  some  permanently 
bound  blank  book,  so  that  reference  may  always  be  had  to  them  at 
any  future  time  for  the  purpose  of  checking  or  reviewing.  Put 
all  the  data  down.  Do  not  leave  in  doubt  the  exact  conditions 
under  which  the  calculations  were  made.  .  Note  the  date  of  calcu- 
lation. 

If  a  mistake  in  figures  is  made,  or  a  change  is  found  neces- 
sary, never  rub  out  the  figures  or  tear  out  the  leaf,  or  in  any  way 
obliterate  the  figures.  Simply  draw  a  bold  cross  through  the  wrong 
part  and  begin  again.  Often  a  calculation  which  is  supposed  to 
be  wrong  is  later  shown  to  be  right,  or  the  facts  which  caused  the 
error  may  be  needed  for  investigation  and  comparison.  Time  which 


MACHINE  DESIGN  9 

is  spent  in  making  figures  is  always  valuable  time,  time  too  pre- 
cious to  be  thrown  away  by  destroying  the  record. 

The  recording  of  calculations  in  a  permanent  .formr-as-  just 
described,  is  the  general  practice  in  all  modern  engineering  offices. 
This  plan  has  been  established  purely  as  a  business  policy.  In 
case  of  error  it  locates  responsibility  and  settles  dispute.  Con- 
sistent designing  is  made  possible  through  the  records  of  past 
designs.  Proposals,  estimates,  and  bids  may  often  be  made 
instantly,  on  the  basis  of  what  these  record  books  show  of  sizes 
and  weights.  This  bookkeeping  of  calculations  is  as  important  a 
factor  of  systematic  engineering  as  bookkeeping  of  business 
accounts  is  of  financial  success. 

The  student  should  procure  for  this  purpose  a  good  blank  book 
with  a  firm  binding,  size  of  page  not  smaller  than  6  by  8  inches 
(perhaps  8  by  11  inches  may  be  better),  and  every  calculation,  how- 
ever small  and  apparently  unimportant,  should  be  made  in  it. 

Sample  pages  of.  engineering  calculations  are  reproduced  in 
Figs.  3  to  9.  Note  the  sketch  showing  the  forces.  Note  the  clear 

c5  .  O 

statement  of  data.  Note  the  systematic  writing  of  the  equations, 
and  the  definite  substitutions  therein.  Note  the  heavy  double 
underscoring  of  the  result,  when  obtained.  There  is  nothing  in 
the  whole  process  of  the  calculation  that  cannot  be  reviewed  at 
any  moment  by  anybody,  and  in  the  briefest  time. 

The  development  of  a  personal  note-book  is  of  great  value  to 
the  designer  of  machinery.  The  facts  of  observation  and  experi- 
ence recorded  in  proper  form,  bearing  the  imprint  of  intimate 
personal  contact  with  the  points  recorded,  cannot  be  equalled 
in  value  by  those  of  any  hand  or  reference  book  made  by  another. 
There  is  always  a  flavor  about  a  personal  note-book,  a  sort  of 
guarantee,  which  makes  the  use  of  it  by  its  author  definite  and 
sure. 

The  habit  of  taking  and  recording  notes,  or  even  knowing 
what  notes  to  take,  is  an  art  in  itself,  and  the  student  should 
begin  early  to  make  his  note-book.  Aside  from  the  value  of  the 
notes  themselves  as  a  part  of  his  personal  equipment,  the  facility 
with  which  his  eye  will  be  trained  to  see  and  record  mechanical 
things  will  be  of  great  value  in  all  of  his  study  and  work.  How 
many  men  go  through  a  shop  and  really  see  nothing  of  the  opera- 


10  MACHINE  DESIGN 

tions  going  on  therein,  or,  seeing  them,  remember  nothing  !  An 
engineer,  trained  in  this  respect,  will  to  a  surprising  degree  be 
able  to  retain  and  sketch  little  details  which*  fall  under  his  eye  for 
a  brief  moment  only,  while  he  is  passing  through  a  crowded  shop. 

Some  draftsmen  have  the  habit  of  copying  all  the  standard 
tables  of  the  various  offices  in  which  they  work.  While  these  are 
of  some  value  in  a  few  cases,  yet  this  is  not  what  is  meant  by  a 
good  note-book  in  the  best  sense.  Ideas  make  a  good  note-book, 
not  a  mere  tabulation  of  figures.  If  the  basis  upon  which  stan- 
dards are  founded  can  be  transferred  to  permanent  personal  record, 
or  novel  methods  of  calculation,  or  simple  features  of  construc- 
tion, or  data  of  mechanical  tests,  or  efficient  arrangement  of 
machinery — if  these  can  be  preserved  for  reference,  the  note-book 
will  be  of  greatest  value. 

Whatever  is  noted  down,  make  clear  and  intelligible,  illus- 
trating by  a  sketch  if  possible.  Make  the  note  so  clear  that 
reference  to  it  after  a  long  space  of  years  would  bring  the  whole 
subject  before  the  mind  in  an  instant.  If  this  is  not  done  the 
author  of  the  note  himself  will  not  have  patience  to  dig  out  the 
meaning  when  it  is  needed ;  and  the  note  will  be  of  no  value. 

METHOD  OF   DESIGN. 

The  fundamental  lines  of  thought  and  action  which  every 
designer  follows  in  the  solution  of  any  problem  in  any  class  of 
work  whatsoever,  are  four  in  number.  The  expert  may  carry  all 
these  in  mind  at  the  same  time,  without  definite  separation  into  a 
a  step-by-step  process;  but  the  student  must  master  them  in  their 
proper  sequence,  and  thoroughly  understand  their  application. 
In  these  four  are  concentrated  the  entire  art  of  Machine  Design. 
When  they  have  become  so  familiar  as  to  be  instinctively  applied 
on  any  and  all  occasions,  good  design  is  the  result.  The  only 
other  quality  which  will  facilitate  still  further  the  design  of  good 
machinery  is  experience;  and  that  cannot  be  taught,  it  must  be 
acquired  by  actual  work. 

i.  Analysis  of  Conditions  and  Forces.  First,  take  a  good 
square  look  at  the  problem  to  be  solved.  Study  it  from  all  sides, 
view  it  in  all  lights,  note  the  worst  conditions  which  can  possibly 
exist,  note  the  average  conditions  of  service,  note  any  special  or 
irregular  service  likely  to  be  called  for. 


MACHINE  DESIGN  11 

With  these  conditions  well  in  mind,  make  a  careful  analysis 
of  all  the  forces,  maximum  as  well  as  average,  which  may  be 
brought  into  play.  Make  a  rough  sketch  of  the  piece  under  con- 
sideration, and  put  in  these  forces.  Be  sure  that  these  forces  are 
at  least  approximately  right.  Go  over  the  analysis  carefully 
again  and  again.  Remember  that  time  saved  at  the  beginning 
by  hasty  and  poor  analysis  will  actually  be  time  lost  at  the  end ; 
and  if  the  machine  actually  fails  from  this  reason,  heavy  financial 
loss  in  material  and  labor  will  occur.  Any  haste  toward  com- 
pletion of  the  structure  beyond  the  roughest  outline,  without  this 
careful  study  of  forces,  is  a  blind  leap  in  the  dark,  entirely  un- 
scientific, and  almost  certain  to  result  in  ultimate  failure. 

On  the  other  hand  this  principle  may  be  carried  too  far.  In 
trying  to  make  the  analysis  thorough  and  the  forces  accurate,  it  is 
quite  possible  to  consume  more  than  a  reasonable  amount  of  time. 
Again,  it  is  not  always  easy,  and  frequently  impossible,  to  deter- 
mine  exactly  the  forces  acting  on  a  given  piece.  But  their  nature, 
whether  sudden  or  slowly  applied,  rapid  in  action  or  only  oc- 
curring at  intervals,  and  their  approximate  direction  and  magni- 
tude at  least,  are  always  capable  of  analysis.  There  are  few,  if 
any,  cases  where  close  assumptions  cannot  be  made  on  the  above 
basis  and  the  design  proceeded  with  accordingly.  Hence  the 
danger  of  too  great  refinement  of  analysis  is  simply  to  be  avoided 
by  the  designer's  plain  business  sense. 

The  first  tendency  of  the  student  is  to  pass  over  the  study  of 
the  forces  as  dull  and  dry,  and  attempt  the  design  at  once.  He 
soon  finds  himself  facing  problems  of  which  he  sees  no  possible 
solution,  and  he  bases  his  design  on  pure  guess-work.  This  is 
the  only  solution  possible  from  such  a  point  of  view,  and  is  really 
no  solution  at  all.  A  guess  which  has  some  rational  backing  is 
often  successful ;  but  in  that  case  some  analysis  is  required,  and  it 
is  not  a  pure  guess,  but  falls  under  the  very  principle  we  are 
considering. 

There  is  no  short  cut  to  the  design  of  machine  parts  which 
avoids  this  full  understanding  of  the  forces  that  they  must 
sustain.  The  size  of  a  belt  depends  upon  the  maximum  pull 
upon  it,  and  the  designing  of  belts  is  nothing  but  providing 
sufficient  cross-section  of  leather  to  prevent  the  belt  tearing  under 


12  MACHINE  DESIGN 

the  pull.  Again,  if  pulley  arms  are  not  to  break,  or  shafts  twist 
off,  or  bolts  be  torn  apart,  or  the  teeth  of  gears  fail,  or  keys  and 
pins  shear  off,  we  must  first,  of  course,  find  out  what  forces  exist 
which  are  likely  to  produce  stress  that  may  lead  to  such 
breakage.  We  should  not  guess  at  the  sizes,  and  then  run  the 
machine  to  see  if  breakage  results,  and  then  guess  again.  Ma- 
chines  are  sometimes  built  in  this  way,  but  it  is  an  unreasonable 
and  uncertain  method.  We  must  use  every  effort  to  foresee  the 
stress  which  a  piece  is  liable  to  receive,  before  we  decide  its  size. 
We  must  know  all  the  forces  approximately,  if  not  positively. 
The  analysis  must  be  thorough  enough  to  permit  of  reasonable 
assumption,  if  not  positive  assertion.  It  is  manifestly  impossible 
to  solve  any  problem  until  we  know  exactly  what  the  problem  is; 
and  a  full  analysis  is  the  statement  of  the  problem. 

2.  Theoretical  Design.  After  we  know  by  careful  analysis 
what  stress  the  machine  part  has  to  sustain,  the  next  step  is  so  to 
design  it  that  it  will  theoretically  resist  the  applied  forces  with 
the  least  expenditure  of  material. 

We  often  see  machinery  with  the  metal  of  which  it  is  made 
distributed  in  the  worst  possible  manner.  In  places  where  the 
stress  is  heavy  and  a  rigid  member  is 'needed,  we  find  a  weak, 
springy  part;  wl\ile  in  other  parts,  where  there  are  no  forces  to  be 
resisted,  or  \  ibration  to  be  absorbed,  there  seems  to  be  a  waste  of 
good  material.  Whether  in  such  case,  the  analysis  of  the  forces 
was  poor,  or  perhaps  not  made  at  all,  or  whether  a  knowledge  of 
how  to  design  so  as  to  resist  the  given  forces  was  wholly  absent, 
cannot  be  told.  At  any  rate,  lack  of  either  or  both  is  clearly 
shown  in  the  result. 

Any  member  of  a  machine  may  vary  in  form  from  a  solid 
block  or  chunk  of  material  to  an  open  ribbed  structure.  The  solid 
chunk  fills  the  requirement  as  far  as  strength  is  concerned,  unless 
it  is  so  heavy  as  to  fail  from  its  own  weight.  But  such  construe- 
tion  is  poor  design,  except  in  cases  where  the  concentration  of 
heavy  mass  ta  necessary  to  absorb  repeated  blows  like  those  of  a 
hammer.  The  possibility  of  these  blows  should,  however,  have 
been  determined  in  the  analysis;  and  the  solid,  anvil  construction 
then  becomes  theoretical  design  for  that  analysis. 

For  steadily  applied  loads  an  open,  ribbed,  or  hollow  box 


MACHINE  DESIGN  13 

structure  can  be  made  which  will  distribute  the  metal  where  it  is 
theoretically  needed,  and  each  fiber  will  then  sustain  its  proper 
share  of  the  load.  In  this  way  weight,  cost,  and  appearance  are 
heeded;  and  the  service  of  the  piece  is  as  good  as,  and  probably 
better  than,  it  would  be  with  the  clumsy,  solid  form. 

There  is  no  such  thing  as  putting  too  much  theory  into  the 
design  of  machinery.  The  strongest  trait  which  an  engineer  can 
have  is  absolute  faith  in  his  analysis  and  calculations,  and  their 
reproduction  in  his  theoretical  design.  Theoretical  design  is  an 
indication  of  scientific  advance  in  the  art,  and  some  of  the  greatest 
steps  of  progress  which  have  been  made  in  recent  years  have  been 
accomplished  through  a  purely  theoretical  study  of  machine 
structure. 

It  will  never  do,  however,  to  be  satisfied  with  theoretical 
design  when  it  is  not  in  accord  with  modern  commercial  and  manu- 
facturing considerations.  Hence  the  next  step  after  the  determina- 
tion of  the  theoretical  design  is  the  study  of  it  from  the  producing 
standpoint. 

3.  Practical  Modification.  All  theoretical  design  viewed  from 
the  business  standpoint  is  worthless,  unless  it  has  been  subjected 
to  the  test  of  cheap  and  efficient  production.  Each  machine  detail, 
though 'correct  in  theory,  may  yet  be  improperly  shaped  and  unfit 
for  the  part  it  is  to  play  in  the  general  scheme  of  manufacture. 

The  conditions  here  involved  are  changeable.  What  is  good 
design  in  this  decade  may  be  bad  in  the  next.  In  this  light  the 
designer  must  be  a  close  student  of  the  signs  of  the  times ;  he  must 
follow  the  march  of  progress,  closely  applying  existing  resources, 
conditions,  and  facilities,  otherwise  he  cannot  produce  up-to-date 
designs.  The  introduction  of  new  raw  materials,  the  cheapening 
of  production  of  others,  the  changing  of  shop  methods,  the  use  of 
special  machinery,  the  opening  of  new  markets,  the  development 
of  new  motive  agents,  —  all  these  and  many  others  are  constantly 
demanding  some  modification  in  design  to  meet  competition. 

Illustrative  of  this,  note  the  change  which  has  been  wrought 
by  the  development  of  electric  power,  the  rise  and  decline  of  the 
bicycle  business,  the  present  manufacture  of  automobiles,  the  last 
named  especially  with  reference  to  the  development  of  the  small 
motive  unit,  the  gasolene  engine,  the  steam  engine,  etc.  The 


14  MACHINE  DESIGN 

design  of  much  machinery  has  been  materially  changed  to  meet 
the  exacting  demands  of  these  new  enterprises. 

Practical  modifications  of  design  necessary  to  meet  the  limi- 
tations of  construction  in  the  pattern  shop,  foundry,  and  machine 
shop  are  of  daily  application  in  the  designer's  work.  He  must 
keep  in  his  mind's  eye  at  all  times  the  workmen  and  the  processes 
they  use  to  create  his  designs  in  metal  in  the  shop. 

"How  can  this  be  made?"  "Can  it  be  made  at  all?" 
"  Can  it  be  made  cheaply  ? "  "  Will  it  be  simple  in  operation 
after  it  is  made  ? "  "  Can  it  be  readily  removed  for  repair  ? " 
"  Can  it  be  lubricated  ? "  "  How  can  it  be  put  in  place  ? "  "  How 
can  it  be  gotten  out  ? "  "  Will  it  be  made  in  small  quantities 
or  large  ? "  "  Will  it  sell  as  a  special  or  standard  machine  ? " 
etc.,  etc. 

The  consideration  of  such  questions  as  these  is  a  practical 
necessity  as  a  business  matter.  No  other  feature  affects  the 
design  of  machinery  more,  perhaps;  for  designs  which  cannot  be 
built  as  business  propositions  are  no  designs  at  all. 

The  student,  it  is  true,  may  not  have  the  extended  shop 
knowledge  which  is  essential  to  this;  but  he  can  do  much  for 
himselt  by  visiting  shops  whenever  possible,  getting  hold  of  shop 
ways  of  doing  things,  and  invariably  treating  his  work  as  a 
business  matter.  Though  a  man  may  not  be  a  pattern  maker, 
molder,  blacksmith,  or  machinist,  yet  he  can  soon  gain  ideas  of  the 
processes  in  each  of  these  branches  which  will  be  of  immense 
advantage  to  him  in  his  designing  work. 

4.  Delineation  and  Specification.  This  means  the  clear  and 
concise  representation  of  the  design  by  mechanical  drawings. 

This  is  as  much  a  part  of  the  routine  method  of  Machine  De- 
sign as  the  other  three  points  which  have  been  discussed.  The 
mere  act  of  putting  the  results  of  mechanical  thinking  on  paper  is 
one  of  the  greatest  helps  to  force  thinking  machinery  to  system- 
atic and  definite  action.  A  designer  never  thinks  very  long 
without  'drawing  something,  and  the  student  must  bring  himself  to 
-  -feel  that  a  drawing  in  its  first  sense  is  a  means  of  helping  his  own 
thought,  and  must  freely  use  it  as  such. 

In  its  second  and  final  sense,  the  drawing  is  an  order  and 
specification  sheet  from  the  designer  to  the  workman.  Design 


MACHINE  DESIGN  15 

which  stops  short  of  exact,  finished  delineation  in  the  form  of 
working  shop  drawings  is  only  half  done.  In  fact  the  possibility 
of  a  piece  being  thus  exactly  drawn  is  often  the  crucial~fest  of  its 
feasibility  as  a  part  of  a  machine.  It  is  easy  to  make  general  out- 
lines, but  it  is  not  so  easy  to  get  down  to  finished  detail.  It  is 
safe  to  say  that  there  is  no  one  thing  productive  of  more  trouble, 
delay  and  embarrassment,  and  waste  of  time  and  money  in  the 
shop,  when  there  need  be  none  from  this  cause,  than  a  poor  detail 
drawing.  The  efficiency  of  the  process  of  design  is  not  fully  real- 
ized, and  failures  are  often  recorded  where  there  should  be  success, 
merely  because  the  indefiniteness  permitted  by  the  designer  in  the 
drawings  naturally  transmitted  itself  to  the  workman,  and  he  in 
turn  produced  a  part  indefinite  in  form  and  operation. 

The  actual  process  of  drawing  in  the  development  of  a  design 
may  be  outlined  as  follows  : 

Hough  sketches  merely  representing  ideas,  not  drawn  to  scale, 
are  first  made.  These  are  of  use  only  so  far  as  the  choice  of  me- 
chanical ideas  is  concerned,  and  to  carry  preliminary  dimensions. 

Following  these  sketches,  comes  a  layout  to  scale,  of  the 
favored  sketch,  a  working  out  of  the  relative  sizes  and  location  of 
the  parts.  This  drawing  may  be  of  a  sketchy  nature,  carrying  a 
principal  dimension  here  and  there  to  fix  and  control  the  detailed 
design.  In  this  drawing  the  design  is  developed  and  general  detail 
worked  out.  The  minute  detail  of  the  individual  parts  is,  1  ^wever, 
left  to  the  subsequent  working  drawing. 

This  layout  drawing  may  now  be  turned  over  to  an  expert 
draftsman  or  detail  designer,  who  picks  out  each  part,  makes  an 
exact  drawing  of  it,  studying  every  little  detail  of  its  shape,  and 
finally  adds  complete  dimensions  and  specifications  so  that  the 
workman  is  positively  informed  as  to  every  point  of  its  construction. 

General  drawings  and  cross  sections  constitute  the  last  step 
in  the  process  of  complete  delineation.  These  show  the  parts 
assembled  in  the  complete  machine.  They  also  serve  a  valuable 
purpose  to  the  draftsman  in  checking  up  the  dimensions  of  the 
detail  drawings.  Errors  -which  have  escaped  previous  notice  are 
often  discovered  in  this  way.  The  layout,  mentioned  above,  is 
sometimes  finished  up  into  a  general  drawing;  but  it  is  safer  to 
make  an  entirely  new  drawing,  as  changes  in  detail  are  often 
necessary  after  the  layout  is  made. 


16  MACHINE  DESIGN 

The  four  fundamental  lines  of  thought  and  action  noted 
above  may  be  summarized  thus — "analyze  and  theorize,  modify 
and  delineate."  This  is  a  maxim  easy  to  remember,  applicable 
to  every  problem  in  Machine  Design,  and  always  provides  the 
answer  to  the  question  "What  shall  I  do,  how  shall  I  proceed ? " 
by  pointing  out  the  proper  sequence  in  the  course  to  be  followed. 

CONSTRUCTIVE   MECHANICS. 

Mechanics  is  a  constructive  science,  its  principles  lying  at  the 
root  of  the  design  and  operation  of  all  machinery.  It  is  usually 
taught,  however,  as  an  advanced  mathematical  subject;  and  the 
student  gets  his  original  conceptions  of  forces,  moments,  and 
beams  in  the  abstract,  before  he  realizes  the  constructive  value  of 
such  conceptions.  By  "  Constructive  Mechanics "  is  meant  the 
study  of  a  machine  purely  from  its  constructive  side,  the  viewing 
of  the  parts  with  respect  to  their  "  mechanics,"  and  satisfying  the 
requirements  of  the  same  in  form  and  arrangement. 

The  student  may  cultivate  this  habit  of  clear,  mechanical  per- 
ception  by  constantly  noting  the  "mechanics"  of  the  simple 
structures  which  he  sees  in  his  daily  routine  of  work.  Aside 
from  machinery,  in  which  the  "mechanics"  is  often  obscure, 
the  world  is  full  of  simple  examples  of  natural  strength  and 
symmetry,  explainable  by  application  of  the  principles  of  pure 
"  mechanics." 

Posts  and  pillars  are  largest  at  their  bases;  overhanging 
brackets  or  arms  are  spread  out  at  the  fastening  to  the  wall; 
heavy  swinging  gates  are  counter-balanced  by  a  ponderous  weight; 
the  old-fashioned  well  sweep  carries  its  tray  of  stones  at  the  end, 
adjusting  the  balance  to  a  nicety;  these  are  examples  of  things 
depending  for  their  form  and  operation  upon  the  principles  of 
"mechanics."  The  building  of  them  involved  "constructive 
mechanics,"  and  yet  their  constructor  perhaps  never  heard  of  the 
scien.ce,  using  merely  his  natural  sense  of  mechanical  fitness 
Such  simple  reasoning  is,  however,  Constructive  Mechanics. 

Forces,  Moments,  and  Beams.  Machines  are  nothing  but  a 
collection  of  (1)  parts  taking  direct  stress,  or  (2)  parts  acting  as 
loaded  beams.  Forces  acting  without  leverage  produce  direct 
stress  on  the  sustaining  part.  Forces  acting  with  leverage  pro- 


MACHINE  DESIGN  17 

duce  a  moment;  the  sustaining  member  is  a  beam,  and  the  stress 
therein  depends  on  the  theory  of  beams,  as  explained  in  "  Me- 
chanics." 

An  example  of  the  first  is  the  load  on  a  rope,  the  force  acting 
without  leverage,  and  the  rope  therefore  having  a  direct  stress  put 
upon  it. 

An  example  of  the  second  is  a  push  of  the  hand  on  the  crank 
of  a  grindstone.  A  moment  is  produced  about  the  hub  of  the 
crank ;  the  arm  of  the  crank  is  a  beam,  and  the  stress  at  any  point 
of  it  may  be  found  by  the  method  of  theory  of  beams. 

Tension,  Compression,  and  Torsion.  The  stress  induced  in 
the  sustaining  part,  whether  tensile,  compressive,  or  torsional,  is 
caused  by  the  application  of  forces,  either  acting  directly  without 
leverage,  or  with  leverage  in  the  production  of  moments. 

The  forces  applied  from  external  sources  are  at  constant  war 
with  th©  resisting  forces  due  to  the  strength  of  the  fibres  of  the 
material  composing  the  machine  members.  The  moments  of  the 
external  forces  are  constantly  exerted  against  and  balanced  by  the 
moments  of  the  internal  resistance  of  the  material.  Hence, 
design,  from  a  strength  standpoint,  is  merely  a  balancing  of 
internal  strength  against  external  force.  In  other  words,  we  may 
in  all  cases  write  a  sign  of  equality,  place  the  applied  effort  on 
one  side,  the  effective  resistance  on  the  other,  and  we  shall  have 
an  equation,  which,  if  capable  of  solution,  will  give  the  proper 
proportions  of  the  parts  considered. 

External  Force  =  Internal  Resistance. 
External  Moment  =  Internal  Moment  of  Resistance 
Expressed  in  terms  of  the  "  Mechanics :" 

P=AS          (,) 

B  or  T=E         (2) 

In  these  formulas,  which  are  perfectly  general, 

P=direct  load  in  pounds. 

A=area  of  effective  material,  in  square  inches. 

S=working  fibre  stress  of  the  material  (tensile,  compressive,  or  shear- 
ing), in  pounds  per  squ-are  inch. 

B  or  T=external  moment  (bending  or  torsional),  in  inch-pounds. 

I=moment  of  inertia  (direct  or  polar),  of  the  resisting  section. 

c= distance  of  the  most  remote  fibre  of  the  resisting  section  from  the 
neutral  axis. 


18  MACHINE  DESIGN 

P  may  produce  direct  tensile,  compressive,  or  shearing  stress. 

B  may  produce  tensile  or  compressive  stress,  and  requires  use  of  direct 
moment  of  inertia  in  either  case. 

T  produces  shearing  stress,  and  requires  use  of  polar  moment  of 
inertia. 

The  origin  of  formula  (1)  is  obvious,  the  assumption  being 
that  the  fibre  stress  is  equally  distributed  to  every  particle  in  the 
area  "A." 

The  development  of  formula  (2)  is  given  in  any  text-book  in 
Mechanics.  It  requires  the  aid  of  the  Calculus,  however.  Any 
good  handbook  gives  values  for  both  the  direct  moment  of  inertia 
and  the  polar  moment  of  inertia  for  quite  a  large  variety  of  sections, 
so  that  further  reference  is  an  easy  matter  for  the  student.  These 
values  are  also  obtained  through  the  methods  of  the  Calculus. 

The  reason  for  introducing  these  formulas  at  this  time  is  to 
call  the  attention  of  the  student  especially  to  the  fact  of  their 
universal  and  fundamental  use  in  all  problems  concerning  the 
strength  of  machine  parts.  Nearly  every  computation  may  be 
reduced  to  or  expanded  from  these  two  simple  equations.  Many 
complex  combinations  occur,  of  course,  which  will  not  permit  sim- 
ple and  direct  application  of  these  formulas,  but  the  student  will 
do  well  to  place  himself  in  perfect  command  of  these  two.  Assuming 
that  he  is  able  to  analyze  forces,  and  compute  the  simple  moment 
at  the  point  where  he  wishes  to  find  the  strength  of  section,  the 
rest  is  the  mere  insertion  of  the  assumed  working  fibre  stress  of 
the  material  in  the  formula  (2)  above,  and  solution  for  the  quantity 
desired. 

When  the  case  is  one  of  combined  stress,  the  relation  becomes 
more  complicated  and  difficult  of  analysis  and  solution.  The  most 
common  case  is  where  bending  is  combined  with  torsion,  as  in  the 
case  of  a  shaft  transmitting  power,  and  at  the  same  time  loaded 
transversely  between  bearings.  In  fact  there  are  very  few  cases  of 
shafts  in  machines,  which,  at  some  part  of  their  length,  do  not 
have  this  combined  stress.  In  this  case  the  method  of  procedure 
is  to  find  the  simple  bending  moment  and  the  simple  torsional 
moment  separately,  in  the  ordinary  way.  Then  the  theory  of 
elasticity  furnishes  us  with  a  formula  for  an  equivalent  bending 
or  an  equivalent  torsional  moment  which  is  supposed  to  produce 
the  same  effect  upon  the  fibres  of  the  material  as  the  combined 


MACHINE  DESIGN  19 

action  of  the  two  simple  moments  acting  together.  In  other 
words,  the  separate  moments  combined  in  action,  being  impossible 
of  solution  in  that  form,  are  reduced  to  an  equivalent~simple 
moment  and  the  solution  then  becomes  the  same  as  for  the  prev- 
ious case. 

These  equivalent  equations  are  given  below,  the  subscript  "e" 
being  added  to  express  separation  from  the  simple  moment: 

Be=-  v+ii/^TT2        (3) 


(4) 

Be  and  Te,  found  from  these  equations,  are  the  external  mo- 
ments, and  are  to  be  equated  to  the  internal  moments  of  resistance 
of  the  section  precisely  as  if  they  were  simple  bending  or  torsional 
moments.  Either  may  be  used.  For  shafts  (4)  is  generally  used, 
being  the  simpler  of  the  two  in  form. 

FRICTION   AND   LUBRICATION. 

The  parts  of  a  machine  which  have  no  relative  motion  with 
regard  to  each  other  are  not  dependent  upon  lubrication  of  their 
surfaces  for  the  proper  performance  of  their  functions.  In  cases 
where  relative  motion  does  occur,  as  between  a  planer  bed  and  its 
ways,  a  shaft  and  its  bearing,  or  a  driving  screw  and  its  nut, 
friction,  and  consequent  resistance  to  motion,  will  inevitably 
Occur.  Heat  will  be  generated,  and  cutting  or  scoring  of  the 
surfaces  will  take  place  if  the  surfaces  are  allowed  to  run  together 
dry. 

This  difficulty,  which  exists  with  all-  materials,  cannot  be 
overcome,  for  it  is  a  result  of  roughness  of  surface,  characteristic 
of  the  material  even  when  highly  finished.  The  problem  of  the 
designer,  then,  is  to  take  conditions  as  he  finds  them,  and,  as  he 
cannot  change  the  physical  characteristics  of  materials,  so  choose 
those  which  are  to  rub  together  in  the  operation  of  the  machine 
that  friction  will  be  reduced  to  the  lowest  possible  limit.  Now  it 
fortunately  happens  that  there  are  certain  agents  like  oil  and 
graphite,  which  seem  to  fill  up  the  hollows  in  the  surface  of  a 
solid  material,  and  which  themselves  have  very  little  friction  on 
other  substances.  Hence,  if  a  machine  permits  by  its  design  an 
automatic  supply  of  these  lubricating  agents  to  all  surfaces  having 


MACHINE  DESIGN 


motion  between  them,  friction  may  be  reduced  to  the  lowest  limit. 

If  this  full  supply  of  lubricant  lie  secured,  and  the  parts  still 
heat  and  cut,  then  the  fault  may  be  traced  to  other  causes,  such  as 
springy  surfaces,  localization  of  pressure,  or  insufficient  radiating 
surface  to  carry  away  the  heat  of  friction  as  fast  as  it  is  generated. 

Lubricating  agents  are  of  a  nature  running  from  the  solid 
graphite  form  to  a  thick  grease,  then  to  a  heavy  dark  oil,  and 
finally  to  a  thin,  fluid  oil  flowing  as  freely  as  water.  The  solid  and 
heavy  lubricants  are  applicable  to  heavily  loaded  places  where  the 
pressure  would  squeeze  out  the  lighter  oils.  Grease,  forced  be- 
tween the  surfaces  by  compression  grease  cups,  is  an  admirable 
lubricator  for  heavy  machinery  under  severe  service.  High-speed 
and  accurate  machinery,  lightly  loaded,  requires  a  thin  oil,  as  the 
fits  would  not  allow  room  for  the  heavier  lubricants  to  find  their 
way  to  the  desired  spot.  The  ideal  condition  in  any  case  is  to 
have  a  film  of  lubricant  always  between  the  surfaces  in  contact, 
and  it  is  this  condition  at  which  the  designer  is  always  aiming  in 
his  lubricating  devices. 

Oil  ways  and  channels  should  be  direct,  ample  in  size. 
readily  accessible  for  cleaning,  and  distributing  the  oil  by  natural 
flow  over  the  full  extent  of  the  surface.  Hidden  and  remote 
bearings  must  be  reached  by  pipes,  the  mouths  of  which  should 
be  clearly  indicated  and  accessible  to  the  operator  of  the  machine. 
Such  pipes  must  be  straight,  if  possible,  and  readily  cleaned. 

There  is  one  practical  principle  affecting  the  design  of 
methods  of  lubrication  of  a  machine  which  should  be  borne  in 
mind.  This  is,  "  Neglect  and  carelessness  by  the  operator  must 
be  provided  for."  It  is  of  no  use  to  say  that  the  ruination  of  a 
surface  or  hidden  bearing  is  due  to  neglect  by  the  operator,  if  the 
means  for  such  lubrication  are  not  perfectly  obvious.  This  is 
"  locking  the  door  after  the  horse  is  stolen."  The  designer  has 
not  done  his  duty  until  he  has  made  the  scheme  of  lubrication  so 
plain  that  every  part  must  receive  its  proper  supply  of  oil,  except 
by  gross  and  willful  negligence,  for  which  there  can  be  no 
possible  just  excuse. 

WORKING   STRESSES   AND   STRAINS. 

Some  persons  object  to  the  use  of  these  terms,  as  one  is 
*ra<pently  used  for  the  other,  and  misunderstanding  results.  This 


MACHINE  DESIGN  21 

is  doubtless  true;  but  the  student  may  as  well  learn  the  true 
relation  of  the  terms  once  for  all,  because  he  will  frequently  run 
across  them  in  his  reading  and  reference  work,  and  should  Intei- 
pret  them  rightly.  The  strict  relation  of  the  two  is  as  follows: 

Stress  is  the  internal  force  in  a  piece  resisting  the  external 
force  applied  to  it.  A  weight  of  ten  pounds  hanging  on  a  rope 
produces  a  stress  of  ten  pounds  in  the  rope. 

Strain  is  the  change  of  shape,  or  deformation,  in  a  piece 
resisting  an  external  force  applied  to  it.  If  the  above  weight  of 
tea  pounds  stretches  the  rope  J  inch,  the  strain  is  J  inch. 

Unit  stress  is  stress  per  unit  area,  e.  g.,  per  square  inch. 

Unit  strain  is  strain  per  unit  length,  e.  g.,  per  inch  length. 

In  the  above  case,  if  the  rope  were  -J  square  inch  in  area 
and  30  inches  long,  the  unit  stress,  or  intensity  of  stress,  is 
10-*-|=20  pounds  per  square  inch;  the  unit  strain  is  i-r-30=T£1f 
inch  per  inch. 

When  stress  is  induced  in  a  piece,  the  strain  is  practically 
proportional  to  the  stress  for  all  values  of  the  stress  below  the 
elastic  limit  of  the  material;  and  when  the  external  load  is  re- 
moved the  strain  will  entirely  disappear,  or  the  recovering  power 
of  the  material  will  restore  the  piece  to  the  original  length. 

Illustrating  by  the  case  above,  on  the  supposition  that  the 
elastic  limit  has  not  been  reached  by  the  stress  of  20  pounds  per 
square  inch,  if  the  load  of  10  pounds  were  taken  off,  the  J-inch 
strain  would  disappear  and  the  rope  return  to  its  original  length; 
if  the  load  were  changed  to  -J  of  10  pounds,  or  5  pounds,  the 
strain  would  be  ^  of  J  inch,  or  J  inch. 

Now  it  is  found  that  if  we  wish  a  piece  to  last  in  service  for 
a  long  time  without  danger  of  breakage,  we  must  not  permit  it 
to  be  stressed  anywhere  near  the  elastic  limit  value.  If  we  do, 
although  it  will  probably  not  break  at  once,  it  is  in  a  dangerous 
condition,  and  not  well  suited  to  its  requirements  as  a  machine 
member.  The  technical  name  for  this  weakening  effect  is  "  fa- 
tigue."  It  is  further  found  that  the  fatigue  due  to  this  repeated 
stress  is  reached  at  a  lower  limit  when  the  stress  is  alternating  in 
character  than  when  it  is  not.  In  other  words,  if  we  first  pull  on 
a  piece  and  then  push  on  it,  we  shall  first  have  the  piece  ia  tension 
and  then  in  compression;  this  alternation  of  stress  repeated  to 


22  MACHINE  DESIGN 

near  the  elastic  limit  of  the  material  will  fatigue  it,  or  wear  out 
the  fibres,  and  it  will  finally  fail.  If,  however,  we  first  pull  on 
the  piece  with  the  same  force  as  before,  and  then  let  go,  we  shall 
first  have  the  piece  in  tension  and  then  entirely  relieved;  such 
repetition  of  stress  will  finally  "  fatigue "  the  material,  but  not  so 
quickly  as  in  the  first  case.  Experiments  indicate  that  it  may 
take  twice  as  many  applications  in  the  latter  case  as  in  the  former. 

The  working  stress  of  materials  permissible  in  machines  is 
based  on  the  above  facts.  The  breaking  strength  divided  by  a 
liberal  factor  of  safety  will  not  necessarily  give  a  desirable  work- 
ing stress.  The  question  to  be  answered  is,  "  Will  the  assumed 
working  fibre  stress  permit  an  indefinite  number  of  applications 
of  the  load  without  fatiguing  the  material  ? " 

Hence  we  see  that  the  same  material  may  be  safely  used  under 
different  assumptions  of  working  stress.  For  example,  a  rotating 
shaft,  heavily  loaded  between  bearings,  acts  as  a  beam  which  in 
each  revolution  is  having  its  particles  subjected,  first  to  a  maxi- 
mum tensile  stress,  and  then  to  a  maximum  compressive  stress. 
This  is  obviously  a  very  different  stress  from  that  which  the  same 
piece  would  receive  if  it  were  a  pin  in  a  bridge  truss.  In  the 
former  we  have  a  case  where  the  stress  on  each  particle  reverses  at 
each  revolution,  while  in  the  latter  we  have  merely  the  same  stress 
recurring  at  intervals,  but  never  becoming  of  the  opposite  char- 
acter. For  ordinary  steel,  a  value  of  8,000  would  be  reasonable  in 
the  former  case,  while  in  the  latter  it  may  be  much  higher  with 
safety,  perhaps  nearly  double. 

From  the  facts  stated  above,  it  is  evident  tHat  exact  values  for 
working  fibre  stress  cannot  be  assumed  with  certainty  and  applied 
broadly  in  all  cases.  If  the  elastic  limit  of  the  material  is  defi- 
nitely known  we  can  base  our  working  value  quite  surely  on  that. 

With  but  a  general  knowledge  of  the  elastic  limit,  ordinary 
steel  is  good  for  from  12,000  to  15,000  pounds  per  square  inch 
non-reversing  stress,  and  8,000  to  10,000  reversing  stress.  Cast 
iron  is  such  an  uncertain  metal  on  account  of  its  variable  structure 
that  stresses  are  always  kept  low,  say  from  3,000  to  4,000  for  non- 
reversing  stress,  and  1,500  to  2,500  for  reversing  stress. 

With  these  values  as  a  guide,  and  the  special  conditions  con. 
trolling  each  case  carefully  studied,  reasonable  limits  may  be 


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MACHINE  DESIGN  23 

assigned  for  working  stress,  not  only  of  steels,  various  grades  of 
cast  iron,  and  mixtures  of  the  same,  but  of  other  alloys,  brass, 
bronze,  etc.  Gun  metal,  semi-steel,  and  bronze  are  intermediate 
in  strength  between  cast  iron  and  steel.  Data  on  the  strength  of 
materials  are  available  in  any  of  the  handbooks,  and  should  be  con- 
suited  freely  by  the  student.  They  will  be  found  somewhat  con- 
flicting,  but  will  assist  the  judgment  in  coming  to  a  conclusion. 

Application  to  Practical  Case.  In  actual  practice  the  only 
information  which  the  designer  has,  upon  which  to  base  his  design, 
is  the  object  to  be  accomplished.  He  must  choose  or  originate 
suitable  devices,  develop  the  arrangement  of  the  parts,  make  his 
own  assumptions  regarding  the  operation  of  the  machine,  then 
Analyze  and  Theorize^  Modify  and  Delineate  each  detail  as 
he  meets  it. 

This,  it  will  be  found,  is  a  very  different  matter  from  taking 
some  familiar  piece  of  machinery,  such  as  a  pulley,  or  a  shaft,  or 
a  gear,  as  an  isolated  case,  the  load  being  definitely  given,  and 
proceeding  with  the  design.  This  is  easily  done,  but  is  only  half 
the  problem,  for  machine  parts,  such  as  pulleys,  gears,  and  shafts, 
do  not  confront  the  designer  tagged  or  labeled  with  the  conditions 
they  are  to  meet.  -He  is  to  provide  parts  to  meet  the  specific  con- 
ditions,  and  it  is  as  much  a  part  of  his  designing  method  to  know 
how  to  attack  the  design  of  a  machine  as  it  is  to  know  how  to 
design  the  parts  in  detail  after  the  attack  has  reduced  the  members 
to  definitely  loaded  structures.  The  whole  process  must  be  gone 
through,  the  preliminary  sketches,  calculations,  and  layout,  all  of 
which  precede  the  detail  design  and  working  drawings ;  and  no  step 
of  the  process  can  be*  omitted. 

It  is  for  this  reason  that  the  present  case  used  for  illustration 
is  carried  out  quite  thoroughly.  The  student  should  make  himself 
familiar  with  every  step  of  the  designing  method  as  applied  to  this 
simple  case  of  design.  More  complex  problems,  handled  in  the 
same  way,  will  simplify  themselves;  and  when  the  point  is  reached 
where  confidence  exists  to  take  hold  of  the  design  of  any  machine, 
however  unfamiliar  its  object  may  be,  or  however  involved  its 
probable  detail  appears,  the  student  has  become  the  true  designer. 
It  is  the  knowing  how  to  attack  a  problem,  to  start  definite  work 
on  it,  to  go  ahead  boldly,  confident  that  the  method  applied  will 


24 


MACHINE  DESIGN 


produce  results,  that  gives  command  of  the  design  of  machinery 
and  wins  engineering  success. 


The  special  case  which   has   been  chosen  to  illustrate  the 
application  of  the  principles  stated  in  the  foregoing  pages  is  ideal, 


MACHINE  DESIGN 

in  tliat  it  does  not  represent  any  actual  machine  at  present  in 
operation.  Probably  builders  of  hoisting  machinery  have  devices 
which  would  improve  the  machine  as  shown.  In  detail,  -as  well 
as  arrangement,  they  could  doubtless  make  criticism  as  manufac- 
turers. The  arrangement  as  shown  is  merely  intended  to  bring 
out  in  simplest  form  the  common  elements  of  transmission  ma- 
chinery  as  parts  of  some  definite  machine,  instead  of  as  isolated 
details.  The  design  is  one  entirely  possible,  practical,  and  me- 
chanical,  but  special  attention  has  been  paid  to  simplicity  in  order 
to  enable  the  student  to  follow  the  method  closely,  for  the  method 
is  the  chief  thing  for  him  to  acquire. 

The  student  is  expected  to  refer  constantly  to  Part  II  for  a 
more  formal  and  general  discussion  of  the  simple  machine  ele- 
ments involved  in  the  case  considered.  Part  II  is  intended  to  be 
a  simplified  and  condensed  reference  book,  carried  out  in  accord- 
ance with  the  method  of  machine  design  as  specified  in  Part  I. 
The  student  should  not  wait  until  he  has  completed  the  study  of 
this  part  before  taking  up  Part  II,  for  the  latter  is  intended  for 
use  with  the  former  in  the  solution  of  the  problems. 

In  the  case  of  power  transmission  about  to  be  studied,  the 
running,  conversational  method  employed  assumes  that  the  student 
is  in  possession  of  the  matter  in  Part  II  on  the  subject  considered. 
Thus,  in  the  design  of  the  pulley,  reference  to  the  subject  of 
u  Pulleys  "  in  Part  II  is  necessary  to  follow  the  train  of  calcula- 
tion; in  designing  the  gear,  consult  "Gears;"  in  calculating  size 
of  shafts,  see  "  Shafts,"  etc.,  etc. 

Problem.  A  machine  is  to  be  designed  to  be  set  on  the  floor 
of  a  building  to  drive  a  wire  rope  falling  from  the  overhead 
sheaves  of  an  elevator  or  hoist.  "Without  regard  to  details  of  this 
overhead  arrangement,  for  its  design  would  be  a  separate  problem, 
suppose  that  the  data  for  the  rope  are  as  follows: 

Load  on  rope 5,000  pounds. 

Speed  of  rope 150  feet  per  minute. 

Length  of  rope  to  be  reeled  in 200  feet. 

We  shall  further  assume  that  the  driving  power  is  to  be  an 
electric  motor  belted  to  the  machine,  that  the  required  speed 
reduction  can  be  satisfactorily  obtained  by  a  single  pair  of  pulleys 
and  one  pair  of  gears,  and  that  a  plain  band  brake  is  to  be  applied 
to  the  drum. 


26 


MACHINE  DESIGN 


"With  this  data  we  shall  proceed  to  work  out  the  detail  design 
of  the  machine. 

Preliminary  Sketch.  The  first  thing  to  do  is  to  sketch 
roughly  the  proposed  arrangement  of  the  machine. 

This  might  appear  like  Fig.  1  except  that  it  would  have  no 
dimensions  in  addition  to  the  data  given  above.  If  the  scheme 
seems  suitable,  the  next  step  is  to  make  such  preliminary  calcula- 
tions as  will  give  further  data,  exact  or  closely  approximate  sizes, 
to  be  put  at  once  on  the  sketch,  to  outline  the  future  design. 

Rope  and  Drum.     Eeferring  to  tables  of  strength  of  wire  rope 
(Kent's  Pocket  Book  gives  the  manufacturers'  list),  we  find  that 
a  |-inch  cast-steel  rope  will  carry  5,000  pounds  safely,  and  that  the 
proper   size  of  drum  to  avoid 
excessive   bending  of   the  rope 
around  it  is  27  inches  diameter. 
Allowing;  i  inch  between  the 

o  o 

coils  as  the  rope  winds  on  the 
drum,  the  pitch  of  coil  will  be 
|  inch  as  shown  in  sketch,  Fig. 
2.  The  length  of  one  complete 

„      27X3.1416 
coil  is,  practically,  - 


12 


Fig.  2. 
200 


=7.07  feet.    To  provide  for  200  feet  will  require  ^r=28-f  coils. 

To  be  safe,  let  us  provide  for  30  coils,  for  which  a  length  of  drum 
(30  X  |) +|=23^  inches  is  required. 

The  space  for  brake  strap  may  be  assumed  at  5  inches,  and  the 
thickness  to  provide  necessary  strength  determined  later  in  the 
design.  The  frictional  surface  of  the  strap  may  be  of  basswc0^ 
blocks,  say  1J  inches  thick,  screwed  to  the  metal  band.  The 
diameter  of  brake  surface  may  be  28  inches. 

Driving  Gears.  The  size  of  drum  gear  evidently  depends 
upon  the  method  of  fastening  to  the  drum,  and,  other  things  being 
equal,  should  be  kept  as  small  as  possible.  One  way  would  be  to 
key  the  gear  on  the  outside  of  the  drum,  another  to  bolt  the  gear 
to  the  end  of  the  drum.  The  latter  has  the  advantage  that  a 
standard  gear  pattern  can  be  used  with  the  slight  change  of 


MACHINE  DESIGN 


addition  of  bolt  flange  on  the  arms.  This  makes  a  simple,  direct, 
and  strong  drive,  the  bolts  being  in  shear. 

Sketching  this  arrangement  as  the  preferred  one  (Fig.  2A),  it 
is  evident  that  the  diameter  of  the  gear  should  be  at  least  as  large 
as  the  drum  in  order  to  keep  the  tooth  load  down  to  a  reasonable 
figure.  On  the  other  hand,  if  made  too  large,  it  spreads  out  the 
machine  and  destroys  its  compactness.  As  a  diameter  of  36 
inches  is  not  excessive,  let  us  assume  this,  and  see  if  a  desirable 
proportion  of  gear  tooth  can  be  found  to  carry  the  load. 

For  a  pitch  diameter  of  36  inches  there  will  be  a  theoretical 

load  of  ^OQX27  =3,750  pounds  at  the  pitch  line.     But  the  load 
ou 


Fig.  2A. 

on  the  tooth  must  not  only  impart  a  pull  of  5,000  pounds  to  the 
rope,  but  must  overcome  friction  between  the  gear  teeth  in  action, 
also  between  the  drum  shaft  and  its  bearings.  Assuming  the 
efficiency  between  the  rope  and  tooth  load  to  be  95  per  cent,  the 


Q  p7F\n 

net  load,  therefore,  which  the  tooth  must  take  is  —  -  =  3,947, 
4,000  pounds. 


MACHINE  DESIGN 


Assuming  involute  teeth,  and  applying  the  "Lewis"  formula, 
(Part  II,  "Gears"): 

W=sXpX/X2/  W=4,000 

8=6,000 
4,000=6,000  X  p  X  /  X  .116  #=.116  (number  of  teeth 

assumed  at  75) 
P  X/=  6>(X^^°116=5.7  inches  ^circular  pitch 

/=face  of  gear 

Let  /=3  jp  (a  reasonable  proportion  for  machine-cut  teeth). 
Then  3  Xp2=5.7 


_ 
p  =|/  1.9=1578  inches 

The  diametral  pitch  corresponding  to  this  is 

3.1416  _OQQ 

T378" 

which  is  just  between  the  regular  standard  pitches,  2  and  2J,  for 
which  stock  cutters  are  made.  To  be  safe,  let  us  take  the  coarser 
pitch,  which  is  2.  The  circular  pitch  corresponding  to  this  is 

—  -  =  1.57,  and  making  the  face  about  three  times  the  circular 

pitch  gives 

3  X  1.57  =  4.71,  say  4J  inches. 

The  number  of  teeth  in  the  gear  is  then  36  X  2  =  72. 
Referring  to  the  value  assumed  for  the  tooth  factor  in  calculation 
above,  it  is  seen  that  y  was  based  on  75  as  the  number  of  teeth, 
which  is  near  enough  to  72  to  avoid  the  necessity  of  further  check- 
ing the  result. 

The  pinion  to  mesh  with  this  gear  should  be  as  small  as  possi- 
ble in  order  to  get  a  high-speed  ratio  between  pinion  shaft  and 
drum,  otherwise  an  excessive  ratio  will  be  required  in  the  pulleys, 
making  the  large  one  of  inconvenient  size.  Small  pinions  have 
the  teeth  badly  undercut  and  therefore  weak,  13  teeth  being  the 
lowest  limit  usually  considered  desirable,  for  that  reason.  Choos- 

13 
ing  that  number,  we  have  a  pitch  diameter  of  "o"—  6.5  in.,  which 

is  probably  ample  to  take  the  shaft  and  key,  and  still  leave  suf- 
ficient stock  under  the  tooth  for  strength.  If  made  of  cast  iron, 
however,  the  pinion  teeth,  on  account  of  the  low  number,  will  be 
narrower  at  the  root  than  those  of  the  gear  of  72  teeth.  Yet  it 


MACHINE  DESIGN  23 

was  upon  the  basis  of  the  latter  that  the  pitch  was  chosen,  for  it 
will  be  remembered  that  the  value  of  y  in  the  formula  was 
taken  at  .116.  Hence  the  pinion  will  be  weaker  than,  the  gear 
unless  we  make  it  of  stronger  material  than  cast  iron,  of  which 
the  large  gear  is  supposed  to  be  made.  Steel  lends  itself  very 
readily  to  this  requirement;  and  in  practice,  pinions  of  less  than  20 
teeth  are  usually  made  of  this  material,  hence  we  shall  specify  the 
pinion  to  be  of  steel. 

Pulleys.  The  question  now  is  whether  or  not  we  can  get  a 
suitable  ratio  in  the  pulleys  without  making  the  large  one  of  incon- 
venient size,  or  giving  the  motor  too  slow  speed  for  an  economical 
proportion. 

Suppose  we  limit  ourselves  to  a  diameter  of  42  inches  for  the 
large  pulley,  and  try  a  ratio  of  4  to  1  ;  this  will  give  a  diameter 
for  the  small  pulley  of  ^-=10^  inches.  We  shall  then  have 


Total  ratio  between  drum  and  motor  ........    ~"X  4==^ 

13  13 

Rev.  per  min.  of  drum  to  give  150  f.  p.  m.  of 


Rev.  per  min.  of  motor  ......................  22.2  X  21.2=470 

Horse-power  of  motor  at  80  per  cent  efficiency  ^^  X  5>(X^)=30 

«33j(XX)  X    .80 

A  30  H.  P.  motor  running  470  r.  p.  m.  would  be  classed  as  a 
slow  speed  motor  and  would  be  a  heavier  machine  and  cost  more 
than  one  of  higher  speed.  It  will  be  noticed,  however,  that  the 
diameter  of  the  small  pulley  is  already  quite  reduced,  and  it-  is 
hardly  desirable  to  decrease  it  still  further.  Neither  can  we 
increase  the  large  pulley,  as  we  have  already  set  the  limit  at  42 
inches.  Hence,  for  our  present  problem  we  cannot  improve  mat- 
ters much  without  increasing  the  size  of  the  large  gear,  which  is 
undesirable,  or  putting  in  another  pair  of  gears,  which  is  contrary 
to  the  conditions  of  the  problem.  As  such  a  motor  is  perfectly 
reasonable,  we  shall  assume  it  to  be  chosen  for  the  purpose. 

In  commercial  practice  it  would  be  well  to  pick  out  some 
standard  make  of  motor  of  the  required  horse-power,  note  the  speed 
as  specified  by  the  makers,  and  then,  if  possible,  suit  the  ratio  in 
the  machine  to  this  speed.  It  is  always  best  to  nse  standard  ma- 
chinery, if  possible,  both  from  the  standpoint  of  first  cost,  as  well 


30  MACHINE  DESIGN 


&>,/?*  3 


'  .  /^<r  2. 


<f 
&,  ^ 

T_.-^72f, 


A? 


Fig.  a 


MACHINE  DESIGN 


81 


as  easa  of  replacing  worn  parts.  Machinery  ordered  special  is 
expensive  in  first  cost  of  designing,  patterns,  and  tools,  and  extra 
spare  parts  for  emergency  orders  are  not  often  kept  on  hand. 

Tabulation  of  Torsional  Moments.  For  future  reference,  it  is 
desirable  at  this  point  to  tabulate  the  torsional  moment,  or  torque, 
about  each  of  the  three  shaft  axes,  assuming  reasonable  efficiencies 
for  the  various  parts,  as  follows: 

Efficiency  between  drum  and  gear  tooth 95  per  cent 

Efficiency  between  drum  and  pinion  shaft 90  per  cent 

Efficiency  between  drum  and  motor  shaft 80  per  cent 

TABLE    OF  TORSIONAL  MOMENTS. 


Axis. 

Inch  Lbs.  Torque 
at  100  Per  Cent  Efficiency. 

Inch  Lbs.  Torque, 
Efficiency  as  Above. 

Drum      . 

500QX—  =67.500 

6L5M 

Pinion           . 

"2 

27      1  ^ 
5  ooox-^-x—  -  =12  187 

.95 

mwuJBii 

5000X-27-x13X10'5     3047 

.90 
8'°47       3809 

.80       0>yUU 

This  means  that  the  motor  develops  a  torque  of  3,809  inch, 
pounds  delivering  to  pinion  shaft  13,541  inch-pounds,  and  to  drum 
71,052  inch-pounds. 

Width  of  Belt.  The  page  of  calculation  for  belt  width  is  repro- 
duced  in  Fig.  3. 

The  calculation  as  given  is  strictly  scientific,  based  on  the 
working  strength  of  a  cemented  joint  (£=400  Ibs.  per  square  inch). 
This  is  a  favorable  situation  for  the  use  of  a  cemented  joint,  be- 
cause it  is  easy  to  provide  means  of  adjusting  the  belt  tension  by 
placing  the  motor  on  a  sliding  base.  Otherwise  a  laced  joint  could 
be  used,  requiring  relacing  when  the  belt  slackens  through  its' 
stretch  in  service.  Under  the  assumption  that  a  double  laced  belt 
is  used,  the  empirical  formula  below  is  one  often  applied: 

wXV_u>X  1,300 
H'  r—  "540" =     ~~540~ 

This  gives  w=  =12*4  inches  (say  12  inches). 

It  should  be  remembered  that  this  value  is  purely  empirical; 
it  applies  to  a  laced  joint,  and  could  not  be  expected  to  check  the 


32  MACHINE  DESIGN 

value  of  9  inches  obtained  by  the  first  computation  for  a  cemented 
joint.  It  is  fairly  in  proportion.  For  the  quite  definite  service 
required  of  the  belt  in  the  present  case,  the  width  of  9  inches  is 
doubtless  sufficient,  considering  the  cemented  joint. 

Length  of  Bearings.  Considerable  latitude  in  choice  of  length 
of  bearings  is  permissible,  especially  in  such  slow-speed  machinery. 
There  is  probably  little  danger  from  heating,  and  the  question  then 
becomes  one  of  wear.  It  is  better  in  such  cases  as  the  one  in  ques- 
tion, to  choose  boldly  a  length  which  seems  to  be  reasonable  and 
proceed  with  the  design  on  that  basis,  even  if  the  length  be  later 
found  out  of  proportion  to  the  shaft  diameter,  than  to  waste  too 
much  time  in  the  preliminary  calculation  over  the  exact  determina- 
tion of  this  question.  Probably  in  most  cases  of  commercial  prac- 
tice the  existence  of  patterns,  or  some  other  practical  consideration, 
will  decide  the  limits  of  length. 

In  the  present  instance  it  seems  reasonable  that  a  length  of 
6  inches  would  fill  the  requirement  for  the  worst  case,  that  of  .the 
drum  shaft,  and  it  is  obvious  that  the  bearings  for  the  pinion  shaft 
would  naturally  be  of  the  same  length  on  account  of  being  cast  on 
the  same  bracket,  and  faced  at  the  same  setting  of  the  planer  tool. 

Height  of  Centers.  The  large  pulley  should  naturally  swing 
clear  of  the  floor.  This  will  require,  say,  a  total  height  of  23 
inches,  out  which  we  may  take  4  inches  for  the  base,  leaving  19 
inches  as  the  height,  center  of  bearing  to  base  of  bracket. 

Data  on  Sketch.  The  data  as  found  above  should  now  be  put 
on  the  sketch  previously  made;  it  will  then  have  the  appearance 
shown  in  Fig.  1. 

This  sketch  is  now  in  form  to  control  all  the  subsequent  detail 
design,  and  it  is  expected  that  the  figured  dimensions  as  shown  can 
be  maintained.  It  is  impossible  to  predict  this  with  positiveness, 
however,  as  in  the  working  out  of  the  minor  details  certain  changes 
may  be  found  desirable,  when,  of  course,  they  should  be  made. 

The  shaft  sizes  do  not  appear  on  this  sketch,  hence  before 
proceeding  further  the  several  shaft  diameters  must  be  calculated. 

Sizes  of  Shafts.  The  calculations  of  the  shaft  diameters  are 
good  instances  of  systematic  engineering  computations,  hence  the^ 
are  reproduced  in  the  exact  form  in  which  they  were  made.  The 
student  should  learn  a  valuable  lesson  in  making  and  recording 


MACHINE  DESIGN 


83 


calculations  by  following  these  carefully.  Note  that  each  set  of 
figures  is  independent,  both  in  the  statement  of  given  data,  as  well 
as  in  the  actual  computation.  Observe  how  easy  it  would  be  for 
the  author  of  these  figures  or  anyone  else  to  check  them  even  after 


r-j-rs /*73 


"kS    o 


Pig.  4. 

a  long  lapse  of  time.  If  the  machine  should  unexpectedly  fail  ir> 
service  the  figures  are  always  available  to  prove  or  disprove  theor- 
etical weakness.  The  right  triangles  merely  indicate  that  the 
value  of  i/^-RT2  was  found  by  the  graphical  method  suggested  in 


34. 


MACHINE  DESIGN 


Part  II,  "  Shafts,"  the  figures  being  put  on  the  triangle  as  a  sim- 
ple and  direct  way  of  recording  both  process  and  result. 

Attention  is  especially  called  to  the  fact  that  in  the  pinion 
shaft  the  size  is  changed  for  each  piece  upon  the  shaft.     This  is 


Fig.  5. 

done  partly  because  it  is  desired  to  show  the  student  that  the  shaft 
at  each  of  these  points  should  be  theoretically  of  different  size. 
It  is  also  done  because  as  a  practical  feature  of  construction  it  is  a 
good  plan  to  change  the  size  when  the  fit  change*,  partly  for  rea- 
sons  of  production  in  the  shop,  partly  for  ease  in  slipping  pieces 


HYDRAULIC  ACCUMULATOR 

Assembled  Drawing  with  Details  on  the  Same  Sheet 


4Bor  TOM  PLA  TES 

SEMI-CIRCULAR,  3-0"  PAD. 


X  96    ACCUMULA  TOR 

SCALE:- 1"  lj",  3  "*  I  FT. 


HYDRAULIC  ACCUMULATOR 
Detail  Drawing,  to  be  used  with  Figure  on  Opposite  Page 


MACHINE  DESIGN 


35 


freely  endwise  on  the  shaft  until  they  reach  their  proper  fit  arid 
location  in  the  assembling  of  the  machine.  ,_-__ 

This  should  not  be  taken  as  an  absolute  requirement  in  any 
sense.  A  straight  shaft  would  be  satisfactory  in*  the  present  case; 
but  the  shouldered  shaft  is  a  little  better  construction,  in  a  mechan- 
ical sense,  and  does  not  cost  much  more.  Hence  it  is  used.  For 
the  drum  the  straight  shaft  seems  to  answer  the  requirement  well 
enough. 


SLI  -03 


Fig.  6. 

Small  Pulley  Bore.     Fig  4. 

Large  Pulley  Bore.     Fig.  5. 

Bearing  Next  to  Large  Pulley.     Fig.  6, 

The  diameter,  2J-J,  as  calculated,  is  based  on  the  supposition 
that  the  greatest  bending  moment  is  caused  by  the  belt  pull  on  the 
overhanging  pulley,  that  is,  by  the  forces  existing  at  the  left-hand 
side  of  the  center  of  the  bearing. 


86 


MACHINE  DESIGN 


But  the  pinion  tooth  load  produces  a  heavy  bending  on  the 
shaft  in  the  bearing,  the  shaft  in  this  case  acting  as  a  beam  sup- 


^r-o 


^g-y  2./'o3 


Fig.  7. 

ported  at  the  two  bearings  and  having  the  tooth  load  applied  as 
shown.     If  this  latter  effect  be  greater  than  the  former,  that  is,  if 


MACHINE  DESIGN  37 


the  bending  moment  produced  by  the  pinion  tooth  load  be  greater 
than  the  bending  moment  produced  by  the  belt  pull,  then  the  diam- 
eter must  be  increased  to  satisfy  the  latter  case.  As  is  S4jen  by 
the  second  calculation  of  Fig.  6,  this  is  not  the  case,  and  the  diam- 
eter stands  at  2-J-J  as  made. 

Pinion  Bore.  Fig.  7.  The  pinion  being  a  driving  fit  upon 
the  shaft,  reinforces  the  shaft  to  such  an  extent  that  it  is  hardly 
possible  for  the  shaft  to  break  off  very  far  inside  the  face  of  the 
pinion;  but  it  is  quite'  possible  that  the  metal  of  the  pinion  may 
give  enough,  or  be  a  little  free  at  the  ends  of,  the  hole,  so  that  the 
shaft  may  be  broken  off,  say  4-  inch  inside  the  face.  In  this  case, 
it  may  fail  from  the  moment  of  the  force  at  the  left-hand  bearing 
or  of  that  at  the  right.  It  may  fail  then  at  (a)  or  (b),  depending 
on  which  section  has  the  greater  bending  moment.  Trying  both, 
it  is  seen  by  the  calculation  that  the  right-hand  moment  is  the 
controlling  one,  and  it,  therefore,  is  used. 

Shaft  Outside  of  Pinion.  Fig.  8.  As  there  is  no  power 
transmitted  through  this  portion  of  the  shaft,  there  is  no  torsional 
moment  in  it,  and  the  bending  moment  remains  practically  the 
same  as  inside  the  pinion. 

The  size  figures  about  2|f ,  but  since  there  is  no  use  in  turn- 
ing  off  material  just  to  reduce  the  size  to  this,  it  is  well  to  make 
it  2  J,  or  just  smaller  than  the  fit  in  the  pinion. 

Pinion  Shaft  Outer  Bearing.  Fig.  8.  This  diameter,  of 
course,  figures  small,  as  there  is  no  torsion  in  it,  and  the  bending 
moment  is  not  heavy.  The  practical  question  comes  in,  however, 
whether  it  is  advisable  to  make  the  outer  bracket  different  from 
the  inner  one  just  on  account  of  this  bearing.  The  commercial 
answer  to  this  would  probably  be  "  No,"  hence  the  size  as  figured 
next  to  the  pinion  will  be  maintained  (2-1--J-). 

Drum  Shaft.  Fig.  9.  In  this  case,  as  previously  inferred, 
the  simplest  thing  to  do  is  to  use  a  piece  of  straight  cold-rolled 
steel,  and  make  both  bearings  alike,  the  size  being  determined 
according  to  the  worst  case  of  loading  which  can  occur  as  the 
rope  travels  from  end  to  end  of  the  .i-um.  This  case  ia  evi- 
dently when  tne  rope  is  at  the  end  of  its  travel  close  to  the  brake, 
for  at  that  time  both  the  load  on  the  rope  and  the  load  on  the  pinion 
tooth  which  is  driving  it  are  exerted  upward,  and  produce  the 


38  MACHINE  DESIGN 

greatest  reaction  at  the  bearing  next  to  the  gear.     The  analysis  of 
the  forces  for  this  condition  is  shown  in  Fig.  9. 

Other  conditions  of  loading  would  be  when  the  brake  is  on 
and  the  tooth  load  relieved,  but  then  the  resultant  of  the  brake 
strap  tensions  would  be  diagonally  downward  and  would  reduce 


03 


Fig.  8. 

rather  than  add  to  the  rope  load.  Again,  when  the  rope  is  at 
the  end  of  the  drum  farthest  from  the  gear,  the  load  on  it  and 
the  load  on  the  pinion  tooth  are  both  exerted  upward  as  before,  but 
the  reaction  cannot  be  as  jreat  as  in  the  case  of  Fig.  9,  because  the 
tooth  load  is  still  concentrated  at  the  other  end  of  the  shaft  and 
produces  a  relatively  small  reaction  at  the  rope  end 


MACHINE  DESIGN 


39 


Preliminary  Layout.  Fig.  10.  Proceeding  now  with  the  lay- 
out to  scale,  the  detail  of  the  parts  may  be  worked  out  as  com- 
pletely as  the  scale  of  the  drawing  will  permit.  The~wT5r!?non-thig 
drawing  may  be  of  an  unfinished,  sketchy  nature,  but  the  measure- 
ments must  be  exact  as  far  as  they  go,  for  this  drawing  is  to  serve 
as  the  reference  sheet,  from  which  all  future  detail  is  to  be  worked  up. 

In  this  layout  may  be  worked  out  the  sizes  of  the  arms  and 
hubs  of  pulleys  and  gears,  the  proportions  of  the  drum  and  brake 


Fig.  9. 

strap,  and  the  general  dimensions  of  the  side  brackets  and  the 
base.  When  the  detail  becomes  too  fine  to  work  out  to  advan- 
tage on  this  drawing  it  may  be  worked  out  full  size  by  a  separate 
sketch,  or  left  to  be  finished  when  it  is  regularly  detailed.  The 
preliminary  layout,  it  should  be  remembered,  is  a  service  sheet 
only,  a  means  of  carrying  along  the  design,  and  not  intended  for 


Fig.  10. 


MACHINE  DESIGN  41 

a  finished  drawing.  The  moment  that  the  free  use  of  the  layout 
IB  impaired  by  trying  to  make  too  much  of  a  drawing  of  it,  its 
value  is  largely  lost.  A  designer  must  have  some  plac^to  try  out 
his  schemes  and  devices,  and  the  layout  drawing  is  the  place  to  do 
it.  This  drawing  may  be  recurred  to  at  intervals  in  the  progress 
of  the  design,  details  being  filled  in  as  they  are  worked  out,  as 
they  may  control  the  design  of  adjacent  parts. 

As  the  discussion  of  the  design  of  each  of  the  members 
involved  in  the  present  problem  can  be  better  taken  up  in  con- 
nection with  the  detail  drawing  of  each,  it  will  be  given  there, 
rather  than  in  connection  with  the  layout,  although  many  of  the 
proportions  thus  discussed  could  be  worked  out  directly  from  the 
latter. 

Pulleys.  Fig.  11.  The  analysis  of  the  forces  in  the  belt 
gives,  according  to  the  calculation  of  Fig.  3,  a  tension  in  the  tight 
side  of  1,059  pounds,  and  in  the  slack  side  414  pounds.  The 
difference  of  these,  or  1,059 — 414=645  pounds,  is  transmitted  to 
the  pulley  and  produces  the  torque  in  the  shaft.  Of  course  in 
the  small  pulley  the  torque  is  transmitted  from  the  motor  through 
the  pulley  to  the  belt,  but  both  cases  are  the  same  as  far  as  the 
loading  of  the  pulleys  is  concerned. 

The  only  other  force  theoretically  acting  is  the  centrifugal 
force  due  to  the  speed  of  the  pulley.  This  produces  tension  in 
the  rim  and  arms,  but  for  the  low  value  of  1,300  feet  per  minute 
peripheral  velocity  in  this  case  may  be  disregarded. 

Considering  the  arms  as  beams  loaded  at  the  ends,  and  that 
one-half  the  whole  number  of  arms  take  the  load,  and  for  con. 
venience,  figuring  the  size  of  the  arms  at  the  center  of  the  pulley 
gives  the  following  calculation  for  the  large  pulley: 

Let     8=2,500 
"       /^breadth  of  oval 
«    .4&=thickness  of  oval 

=  1^16  =3.6  (say 3.5) 

5=1  A  (say  17-16) 
This  is  about  all  the  theoretical  figuring  necessary  on  this 
pulley.     The  rim  is  made  as*  thin  as  experience  judges  it  capable 
of  being  cast;  the  arms  are  tapered  to  suit  the  eye,  thus  giving 
ample  fastening  to  the  rim  to  provide  against  shearing  off  the  rim 


Fig.  11 


MACHINE  DESIGN  43 

from  the  arms;  generous  fillets  join  the  arms  to  both  rim  and  hub; 
and  the  hub  is  given  thickness  to  carry  the  key,  and  length 
enough  to  prevent  tendency  to  rock  on  the  shaft,  _IJncertain 
strains  due  to  unequal  cooling  in  the  foundry  mold  may  be  set  up 
in  the  arms  and  rim,  but  with  careful  pouring  of  the  metal  they 
should  not  be  serious,  and  the  low  value  chosen  for  the  fibre  stress 
allows  considerable  margin  for  strength. 

The  small  pulley  has  the  same  forces  to  withstand  as  the 
large  pulley,  but  on  account  of  its  small  diameter  there  is  not 
room  enough  for  arms  between  the"  rim  and  the  hub,  hence  it  is 

o 

made  with  a  web.  The  web  cannot  be  given  any  bending  by  the 
belt  pull,  the  only  tendency  which  exists  in  this  case  being  a 
shearing  where  the  web  joins  the  hub.  This  shearing  also  exists 
throughout  the  web  as  well,  but  at  other  points  farther  from  the 
center  it  is  of  less  magnitude,  and  moreover,  there  is  more  area  of 
metal  to  take  it.  The  natural  way  to  proportion  the  thickness  of 
the  web  is  to  give  it  an  intermediate  thickness  between  that  of  the 
hub  and  rim,  thus  securing  uniform  cooling,  and  then  figure  the 
stress  as  a  check.  Making  this  value  g-  inch  gives  a  shearing  area 
of  g  multiplied  by  the  circumference  of  the  hub,  which  is  3.1410 

X  4  =  12.56.     The  shearing  force  at  the  hub  is ^       =1,693 

pounds.     Equating  the  external  force  to  the  internal  resistance 
1,693  =  |  X  12.56XS 

1,693X8       - -,  .         ,  .-.:•.,  , 

~  7x12  56  ~         Pounds  Per  square  inch  (approx.). 

This  is  a  very  low  figure,  even  for  cast  iron,  hence  the  web  is 
amply  strong.  The  rim  and  hub  are  proportioned  as  for  the  large 
pulley. 

The  keys  are  taken  from  the  standard  list.  They  may  be 
checked  for  shear,  crushing"  in  the  hub,  and  crushing  in  the  shaft. 

'  O  '  O 

but  the  hubs  are  so  long  that  it  is  at  once  evident  without  figuring 
that  the  stress  would  run  very  low  in  both  cases. 

Gears.  Fig.  12.  The  analysis  of  the  forces  acting  on  the 
gears  has  been  given  on  page  28,  4,000  pounds  being  taken  at  the 
pitch  line.  Using  this,  same  value,  and  choosing  a  T-shaped 
arm  as  a  good  form  for  a  heavily  loaded  gear  like  the  present  one, 
let  us  consider  that  the  rim  is  stiff  enough  to  distribute  the  load 


Fig.  12. 


MACHINE  DESIGN  45 

equally  between  all  the  arms,  and  that  each  acts  as  a  beam  loaded 
at  the  end  with  its  proportion  of  the  tooth  load.  Before  we  can 
determine  the  length  of  these  arms,  however,  we  must  fix  upon  the 
size  of  the  flange  which  is  to  carry  the  driving  bolts.  This  is  taken 
at  13  inches.  It  could  be  smaller  if  desired,  but  drawing  the  bolts 
in  toward  the  center  increases  the  load  on  them,  and  13  inches 
seems  reasonable  until  it  is  proved  otherwise.  This  makes  the 

4,000X11.5 
maximum  moment  which  can  come  on  an  arm  -  ^  -  r=7?666 

inch-pounds. 

Now  it  is  evident  that  the  base  of  the  T  arm  section,  which 
lies  in  the  plane  of  rotation,  is  most  effective  for  driving,  and 
that  the  center  leg  of  the  T  does  not  add  much  to  the  driving 
capacity  of  the  arm,  although  it  increases  the  lateral  stiffness  of 
the  arm.  as  well  as  providing  in  casting  a  free  flow  of  metal  between 
the  rim  and  the  hub.  Hence  the  simplest  way  of  treating  the  sec- 
tion of  the  arm  for  strength  is  to  consider  the  base  of  the  T 
only,  of  rectangular  section,  breadth  &,  and  depth  A,  for  which 

SX^XA2. 

the  internal  moment  or  resistance  is  --  ~  - 

o 

Also,  it  is  simplest  to  assume  one  dimension,  say  the  breadth, 
and  the  allowable  fibre  stress,  and  figure  for  the  depth.  Taking 
the  breadth  at  1J  inches,  which  looks  about  right,  and  the  fibre 
stress  at  2,500,  and  equating  the  external  moment  to  the  internal, 
we  have 

7  eee  _  2,500  X  1.125  X  A2 

6X7,666 
"2,500X1.125" 


A  =  ^TO  =  4.05  (say  4J) 

Drawing  in  this  size,  and  tapering  the  arm  to  the  rim  as'  in 
the  case  of  the  pulleys,  making  the  depth  of  the  rim  according  to 
the  suggested  proportions  given  in  Part  II,  "  Gears,"  giving  the 
center  leg  of  the  T  a  thickness  of  |-  inch  tapering  to  1  inch,  and 
heavily  filleting  the  arms  to  the  rim  and  center  flange,  we  have  a 
fairly  well  proportioned  gear. 

The  next  thing  to  determine  is  the  size  of  the  driving  bolts. 
The  circle  upon  which  their  centers  lie  may  be  11  inches  in  diam- 


Fig.  13. 


MACHINE  DESIGN  47 


eter,  and  there  will  naturally  be  six  bolts,  one  between  each  arm. 
These  bolts  are  in  pure  shear,  and  the  material  of  which  they  are 
to  be  made  ought  to  be  good  for  at  least  8,000  pounds_pej*  square 
inch  fibre  stress.  The  force  acting  at  the  circumference  of  an 

11-inch  circle  would  be    '  '     g  -  —13,091  pounds. 

5.5  r 

Equating  the  load  on  each  bolt  to  the  resisting  shear  gives 
•     13,091  8,000x3.1416x^2      Let  A  =  area  resisting  shear. 

-6-;=8,OOOxA=         ~T~  Let  d=dia.  of  bolt. 


4X13,091 
'"6X8,000X3.1416"' 


d—  i/lfc  (say  .6)     %-inch  bolts  would  do. 

But  |-inch  bolts  are  pretty  small  to  use  in  connection  with  such 
heavy  machinery.  They  look  out  of  proportion  to  the  adjacent 
parts.  Hence  g-inch  bolts  have  been  substituted  as  being  better 
suited  to  the  place  in  spite  of  the  facf  that  theoretically  they  are 
larger  than  necessary.  The  extra  cost  is  a  small  matter.  These 
bolts  may  crush  in  the  flange  as  well  as  shear  off,  but  as  there  is 

an  area  of  |X  If  =  1.422  square  inches   to  take  —  ^p  —  =2,182 

pounds,  the  pressure  per  square  inch  of  projected  area  is  only 

2  1  82 

1\  5^=1,534  pounds,  which  is  very  low. 

This  gear  needs  no  key  to  the.  shaft  because  all  the  power 
comes  down  the  arms  and  passes  off  to  the  drum  through  the  bolts. 

JT  o  ' 

thus  putting  no  torsional  stress  in  the  shaft.  The  face  of  the 
flange  is  counterbored  so  as  to  center  the  gear  upon  the  drum, 
without  relying  upon  the  fit  of  the  gear  upon  the  shaft  to  do  this. 

The  pinion  is  solid  and  needs  no  discussion  for  its  design. 

Brackets  and  Caps.  Fig.  13.  As  the  size  of  the  drum  shaft 
was  determined  by  considering  the  rope  wound  close  up  to  the 
brake,  thus  giving  in  combination  wkh  the  load  on  the  gear  tooth 
the  maximum  reaction  at  the  bearing  as  6,743  pounds,  the  cap  and 
bolts  should  be  designed  to  carry  the  same  load. 

For  a  bearing  but  6  inches  long,  two  bolts  are  sufficient  under 
ordinary  conditions  and  might  perhaps  do  for  this  case.  The  load 
is  pretty  heavy,  however,  and  it  is  deemed  wise  to  provide  four 
bolts,  thus  securing  extra  rigidity,  and  permitting  the  use  of  bolts 


43  MACHINE  DESIGN 

of  comparatively  small  size.  If  the  load  were  distributed  equally 
over  all  the  bolts  each  would  take  one-fourth  of  the  whole  load, 
but  it  is  not  usually  safe  to  figure  them  on  this  basis,  because  it 
is  difficult  to  guarantee  that  each  bolt  will  receive  its  exact  share 
of  stress.  Assuming  that  the  two  bolts  on  one  side  take  |  the 
whole  load  instead  of  ^,  which  provides  for  this  uncertain  extra 
stress,  each  bolt  must  take  care  of  -i-  of  6,748,  or  2,249,  pounds. 
Allowing  8,000  pounds  per  square  inch  fibre  stress  calls  for  ar 

2  249 

area  at  the  root  of  the  thread  of  ~      -^  =  .281   square  inch.     Con- 

o,UUU 

suiting  a  table  of  bolts  wo  find  that  the  next  standard  size  of  bolt 
greater  than  this  is  |,  which  gives  an  area  of  .302  square  inch. 
v  Choosing  this  size  as  satisfactory,  the  bolts  should  be  located 
as  close  to  the  shaft  as  will  permit  the  hole  to  be  drilled  and  tapped 
witnout  breaking  out.  A  center  distance  of  5J  inches  accomplishes 
this  result.  The  distance  between  centers  in  the  other  direction 
is  somewhat  arbitrary,  although  the  theoretical  distance  between 
the  bolt  and  the  end  of  the  bearing  to  give  equal  bending  moment 
at  the  center  of  the  cap  and  at  the  line  of  the  bolts  is  about  -f^  of 
the  length,  or  •£%  of  6  =  1 J  inches.  This  proportion  answers 
well  for  the  present  case,  although  for  long  caps  it  brings  the 
bolts  too  far  in  to  look  well. 

The  thickness  of  the  cap  may  be  determined  by  assuming  it 
to  be  a  beam  supported  at  the  bolts  and  loaded  at  the  middle. 
This  is  not  strictly  true,  for  the  load  is  distributed  over  at  least  a 
portion  of  the  shaft  diameter;  moreover,  the  bolts  to  some  extent 
make  the  beam  fixed  at  the  ends.  It  being  impossible  to  determine 
the  exact  nature  of  the  loading,  we  may  take  it  as  stated,  supported 
at  the  ends  and  loaded  in  the  middle,  and  allow  a  higher  fibre 
stress  than  usual,  say  3,500.  The  longitudinal  section  at  the 
middle  of  the  cap  is  rectangular,  of  breadth  6  inches,  and  der»th 
unknown,  say  A.  The  equation  of  moments  is 
Wxl _SXI  _SX£XA2 

4  c  6 

6,748  X  5.5  _  3,500  X  6  X  A2 
~T~  ~~6~ 

6X6,748X5.5 
4X3,500X6 
h  =  1/2.65=1.62  (1J  will  probably  answer) 


MACHINE  DESIGN  49 

For  the  other  bearing  next  to  the  pinion,  the  load  on  the  tooth 
acts  downward,  and  the  resultant  pull  of  the  belt  is  nearly  hori- 
zontal, hence  the  cap  and  bolts  must  stand  but  little-load,  and 
calculation  would  give  minute  values.  In  a  case  like  this  it  is 
well  to  make  the  size  the  same  as  for  the  larger  bearing,  unless 
the  construction  becomes  very  clumsy  thereby.  This  saves  chang- 
ing drills  and  taps  in  making  the  holes,  and  preserves  the  symmetry 
of  the  bracket.  The  |-inch  bolts  are  good  proportion  for  the 
smaller  bearing,  hence  that  size  will  be  maintained  throughout. 

The  body  of  the  bracket  is  conveniently  made  with  the  web  at 
the  side  and  horizontal  ribs  extending  to  the  outside.  The  load  due 
to  the  rope  is  carried  directly  down  the  side  ribs  and  web  into 
the  bottom  flanges  and  to  the  bolts.  The  analysis  of  the  forces 
on  these  bolts  is  shown  in  Fig.  14.  It  is  evident  from  the  figure 
that  the  resultant  belt  pull  tends  to  hold  the  bracket  down,  while 
the  load  on  the  rope  tends  to  pull  it  up,  the  point  about  which  it 
tends  to  rotate  being  the  corner  furthest  from  the  drum.  It  is  also 
evident  thac  the  bolts  nearest  this  corner  can  have  little  effect  on 
the  holding  down,  because  their  leverage  is  so  small  about  the  cor- 
ner,  hence  we  shall  assume  that  the  pair  of  bolts  at  the  right-hand 
end  of  the  bracket  takes  all  the  load.  The  belt  pull,  being  hori- 
zontal, tends  to  slide  the  bracket  along  the  base,  but  this  tendency 
is  small,  and  at  any  rate  is  easily  taken  care  of  by  the  two  dowel 
pins,  which  are  thus  put  in  shear. 

The  load  on  the  bolts  being  4,954  pounds,  a  heavy  bending 
moment  is  thrown  on  the  flange  of  the  bracket,  tending  to  break 
it  off  at  the  root  of  the  fillet.  The  distance  to  the  root  of  the  fillet 
is  |  inch;  the  section  tending  to  break  is  rectangular,  of  breadth 
5J  inches,  and  unknown  depth  A.  The  equation  of  moments  is 


- 

c  o 

4,954X3      2,500  X  5.5  xAa 
4  6 

6X4,954X3 

^x^oooxs.s-1'6' 

*=1/1.62==1.3(sayl|). 

The  thickness  of  the  web  and  ribs  of  this  bracket  is  hardly 
capable  of  calculation.     The  figure  |  inch  has  been  chosen  in  pro- 


50 


MACHINE  DESIGN 


portion  to  the  size  of  the  large  drum  bearing,  giving  ample  stiff- 
ness and  rigidity,  and  permitting  uniform  flow  and  cooling  of  the 
metal  in  the  mold.  The  opening  in  the  center  is  made  merely  to 
save  material,  as  in  that  part  little  stress  would  exist,  the  two  sides 


5000 


W  = 


W= 


Fig.  14. 

carrying  the  load  down  to  the  base  bolts,  and  the  top  serving  as  a 
tie  between  the  bearings. 

This  bracket  might  be  made  with  the  web  in  the  center  of  the 
Barings  instead  of  at  the  side,  in  which  case  the  expense  of  the 


MACHINE  DESIGN  51 

pattern  would  be  slightly  greater.  It  could  also  be  made  of  closed 
box  form,  but  would  in  that  case  probably  weigh  more  than  as 
shown. 

Drum  and  Brake.  Fig.  15.  The  analysis  of  the  forces  acting 
on  the  drum  is  simple,  but  its  theoretical  design  is  more  compli- 
cated. It  is  evident  that  the  drum  acts  as  a  beam  of  hollow  circular 
cross  section,  and  that  its  worst  case  of  loading  is  when  the  rope  is 
at  or  near  the  middle  of  the  drum  length.  At  the  same  time  the 
metal  of  this  circular  cross  section  is  in  a  state  of  torsion  between 
the  free  end  of  the  rope  and  the  driving  gear,  due  to  the  load  on 
the  gear  tooth  and  the  reaction  of  the  rope.  Also  the  wrapping  of 
the  rope  around  the  drum  tends  to  crush  the  metal  of  the 
section  beneath:  it,  the  maximum  effect  of  this  action  being  near 
the  free  end  of  the  rope  where  its  tension  has  not  been  reduced  by 
friction  on  the  drum  surface. 

Now  the  "mechanics"  to  solve  the  problem  of  these  three 
combined  actions  is  rather  complicated.  It  can  be  at  least  approx- 
imately solved,  however,  for  it  satisfies  fairly  well  the  case  of 
combined  compression  and  shear.  But  on  a  further  study  of  this 
particular  case,  it  is  seen  at  once  that  the  diameter  of  the  drum  is 
relatively  large  with  respect  to  its  length,  which  means  that  the 
thickness  of  the  metal  may  be  very  small  and  yet  give  a  large 
resisting  area,  or  value  of  "I,"  both  in  direct  bending  as  well  as 
torsion;  also  it  is  so  short  that  the  external  bending  moment  will 
be  small.  The  practical  condition  now  comes  in,  that  the  drum 
can  be  safely  cast  only  when  the  thickness  of  the  metal  is  at  a 
minimum  limit,  for  the  core  may  be  out  of  round,  not  set  centrally, 
or  by  some  other  variation  produce  thin  spots  or  even  develop  holes 
reaching  out  into  the  rope  groove,  discovered  only  when  the  latter 
is  turned  in  the  lathe. 

Hence  it  seems  reasonable  and  safe  in  this  case  to  make  the 
thickness  of  the  drum  depend  simply  upon  the  crushing  caused  by 
the  wrapping  of  the  rope  around  it,  and  we  shall  take  the  coil 
nearest  the  free  end  of  the  rope,  assuming  that  it  carries  the  full 
load  of  5,000  pounds  throughout  one  complete  wrap  around  the 
drum. 

The  area  resisting  the  crushing  action  may  be  considered  to 
be  that  of  the  cross  section  of  a  ring,  of  width  equal  to  the  pitch 


Ffo.  15. 


MACHINE  DESIGN  53 

of  the  groove.  Assuming  that  |  inch  is  the  least  thickness  which 
can  be  safely  allowed  under  the  groove  for  casting  purposes,  let 
us  figure  the  crushing  fibre  stress  to  see  if  this  is  sufficiently 
strong.  Disregarding  the  small  amount  of  metal  existing  above 
the  bottom  of  the  groove,  this  gives  the  area  to  resist  the  crushing 
||Xf=  J-J-,  or  .47  inch.  Since  there  are  two  of  these  sections  and 
the  rope  acts  on  both  sides,  the  equation  of  forces  is: 
5,000X2  =  SX.47X2 

S  =    '     — -    =  10620  pounds  per  square  inch. 

This,  for  cast  iroil,  in  pure  crushing,  allows  plenty  of  margin 
for  the  extra  bending  and  torsional  stress,  which  for  such  a  con- 
siderable thickness  would  be  slight. 

The  above  case  indicates  a  method  of  reasoning  much  used  in 
designing  machinery,  which  while  following  out  the  specified 
routine  of  thought  as  previously  given  in  these  pages,  stops  short 
of  elaborate  and  minute  theoretical  calculation  when  such  is  obvi- 
ously unnecessary.  If  a  drum  of  great  length  were  to  be  designed, 
and  of  small  diameter,  the  same  method  of  reasoning  would  deduce 
the  fact  that  the  design  should  be  based  on  the  bending  and  the 
torsional  moments,  the  thickness  in  such  a  case  being  so  great  to 
withstand  these  that  the  intensity  of  the  crushing  due  to  wrap  of 
the  rope  becomes  of  inappreciable  value. 

The  remaining  points  of  design  of  the  drum  are  determined 
from  practical  considerations  and  judgment  of  appearance.  The 
ribs  behind  the  arms  are  put  in  to  give  lateral  stiffness  and  guard 
against  endwise  collapse.  The  arms  are  subject  to  the  same  bend- 
ing as  those  of  the  gear,  but  as  they  are  equally  heavy  it  is  not 
necessary  to  calculate  them.  The  flange  at  the  driving  end  is  of 
course  matched  to  that  already  designed  for  the  gear.  The  rope 
is  intended  to  be  brought  through  the  right-hand  end  with  an 
easy  bend  and  the  standard  form  of  button  wedged  on  to  prevent 
its  pulling  through. 

This  drum  would  probably  be  cast  with  its  axis  vertical,  and 
the  driving  flange  down  to  secure  sound  metal  at  that  point. 
Heavy  risers  would  be  left  at  the  other  end  to  secure  soundness 
where  the  rope  is  fastened.  Drums  are  often  cast  with  the  axis 
horizontal,  but  the  vertical  method  is  more  certain  to  produce 
a  sound  casting.  The  grooves  should  be  turned  from  thp 


54  MACHINE  DESIGN 

solid  metal,  partly  because  it  is  a  difficult  matter  to  cast  them,  but 
principally  because  the  rope  should  run  on  as  smooth  surface  as 
possible  to  avoid  undue  wear.  On  drums  which  carry  chain  instead 
of  wire  rope  the  grooves  are  sometimes  cast  with  success,  although 
even  in  this  case  the  turned  groove  is  generally  preferable. 

The  brake  consists  of  a  wrought-iron  band  to  which  are  fast- 
ened wooden  .blocks,  the  iron  band  giving  the  requisite  strength 
while  the  blocks  give  frictional  grip  on  the  drum  surface  and  can 
be  easily  replaced  when  -worn.  As  in  the  designing  of  a  belt  the 
object  in  view  is  the  grip  on  the  pulley  surface  by  the  leather  to 
enable  power  to  be  transmitted  from  the  belt  to  the  pulley,  so  in 
the  case  of  the  brake  if  we  put  the  proper  tension  in  the  strap  it 
can  be  made  to  grip  the  brake  drum  so  tightly  that  motion  between 
it  and  the  drum  cannot  occur.  The  latter  case  is  really  the  reverse 
of  the  first,  if  the  driven  pulley  be  considered,  but  is  identical  with 
the  case  of  the  driving  pulley,  in  which  the  power  is  transmitted 
from  the  pulley  to  the  belt.  Of  course  in  the  case  of  the  brake 
no  power  is  transmitted,  as  when  the  brake  holds  no  motion  occurs, 
but  the  principle  of  the  relative  tensions  in  the  strap  is  the  same 
as  for  the  belt. 

Since  the  brake  drum  surface  is  28  inches  in  diameter,  the  load 
at  that  surface  which  the  brake  must  hold  is 

5,000X27 

=  4,821  pounds. 


We  have  then  the  following  calculation  corresponding  exactly 
to  that  of  the  belt  given  in  Fig.  3. 


Tn—  T=P  =  4,821 

.       =  2.729  X  .25  X  .75  .=  0.512  (for  wMch  **  BimbM 


uo 

Then  ?2-=  3.25     T0  =  ^ 


T  --4821  T      -J^-==2'25XTn==4821 

—  J-o-  •  4,8^1  in 


Tn  =    >25>=  6,963  pounds  (say  7,000) 
T0  =  6.963—  4,821  =  2,142  pounds  (say  2,200) 


MACHINE  DESIGN  55 

The  tight  end  of  the  strap  must  then  be  capable  of  carrying  a 
load  of  7,000  pounds,  and  since  the  width  has  already  been  taken  at 
4J  inches,  the  problem  is  to  find  the  necessary  thickness,  ^Equating 
the  external  load  to  the  internal  resistance  we  have 
7,000  =  A  X  S  Let  t  =  thickness 

"  S  =  fibre  stress  =  12,000 
7,000=4.5X2X12,000 


This,  however,  can  be  but  a  preliminary  figure,  for  the  riveting 
of  the  strap  will  take  out  some  of  the  effective  area,  and  the  thick- 
ness will  have  to  be  increased  to  allow  for  this.  Suppose  on  the 
basis  of  this  figure  we  assume  the  thickness  at  a  slightly  increased 
value,  say  T36  inch,  and  proceed  to  calculate  the  rivets. 

A  group  of  five  rivets  will  work  in  well  for  this  case,  which 

7  000 
gives  —  '-=  —  =  1,400  pounds  per  rivet.     A  safe  shearing  fibre  stress 

is  6,000,  hence  the  area  necessary  per  rivet  is  TTTuyj  ==•  -23  square 


inch.  This  comes  nearest  to  the  area  T95  diameter,  but  for  the 
sake  of  using  the  more  general  size  of  rivet  (|  inch)  the  latter  is 
chosen,  for  which  the  area  is  .30. 

We  must  DOW  try  these  rivets  in  a  -^-inch  plate  for  their  safe 
bearing  value.  The  projected  area  of  a  |-inch  hole  in  a  T3g--inch  plate 

is  I  x^  =.117  square  inch.  -pry  =  11,965  (15,000  would  besafe) 

Taking  out  two  |-inch  rivete  from  the  full  width  of  4  J  inches 
leaves  4J  —  (2  X  f  )=  3.25,  and  makes  the  net  area  of  strap  to  take 
stress  3.25  X-^ir=.61  square  inches.    He-calculating  the  fibre  stress 
for  this  area  gives 
7,000==  .61X8 

,  ~  __  7,000  =  11,475  (which  approximates  the  previous   value 
~^T  of  12,000). 

The  slack  end  of  the  strap  has  to  take  but  2,200  pounds,  hence 
a  different  calculation  might  be  made  for  this  end  giving  smaller 
rivets;  but  as  it  is  impractical  to  change  the  thickness  of  the  strap 
to  meet  this  reduced  load,  it  is  well  to  maintain  the  same  proper- 
tion  of  joint  as  at  the  tight  end.  The  spacing  of  th'e  riveta  in  both 


Fig.  18. 


MACHINE  DESIGN  57 

cases  follows  the  ordinary  rule  allowing  at  least  three  times  the 
diameter  of  the  rivet  as  center  distance,  and  one-half  this  value  to 
the  edge  of  the  plate. 

The  threaded  end  of  the  forging  on  the  strap  also  has  to  carry 
the  load  of  2,200  pounds,  for  which  a  size  smaller  than  1  inch 
would  suffice.  It  is  natural,  however,  for  the  sake  of  general  pro- 
portion to  make  the  bolt  as  strong  as  the  strap,  and  a  1-inch  bolt 
gives  an  area  of  .52  square  inch,  nearly  equalling  the  value  of 
.61  net  area  of  strap  noted  above. 

Base,  Brake=Strap  Bracket  and  Foot  Lever.  Fig  16.  Tha 
base  cannot  be  definitely  calculated,  and  can  best  be  proportioned 
by  judgment.  It  must  not  distort,  twist,  or  spring  in  any  way  to 
throw  the  shafts  out  of  line.  The  area  in  contact  with  the  founda- 
tion upon  which  it  rests  must  be  ample  to  carry  the  weight  of  the 
whole  machine  with  a  low  unit  pressure.  Although  the  form 
shown  is  perfectly  practicable  to  cast  and  machine,  and  is  simple 
ard  rigid,  yet  it  is  questionable  if  a  bolted-up  construction,  say  of 
four  pieces,  might  not  be  equally  rigid  and  yet  involve  greater 
facility  of  production  in  both  the  foundry  and  machine  shop  on 
account  of  the  reduced  sizes  of  parts  to  be  handled.  This  is  a 
question  which  depends  on  the  equipment  and  methods  of  the 
individual  shop,  and  is  an  illustration  of  the  practical  control  of 
design  by  manufacturing  conditions. 

The  brake-strap  bracket  and  foot  lever,  also  shown,  in  this 
figure,  are  examples  of  machine  parts  which  are  quite  definitely 
loaded,  and  the  designing  of  which  is  a  simple  matter.  Further 
discussion  of  their  design  is  not  made,  the  student  being  given 
opportunity  for  some  original  thought  in  determining  the  forces 
and  moments  that  control  their  design. 

Gear  Guard  and  Brake=Relief  Spring.  In  exposed  machin- 
ery of  this  character  it  is  desirable  to  cover  over  the  gears  with  a 
guard  to  prevent  anything  accidentally  dropping  between  the 
teeth  and  perhaps  wrecking  the  whole  machine.  This  guard  is  not 
shown,  as  it  involves  little  of  an  engineering  nature  to  interest 
the  student.  It  could  readily  be  made  of  sheet  metal  or  light 
boiler  plate,  bent  to  follow  the  contour  of  the  gears  and  fastened 
to  the  top  flange  of  the  main  bracket. 

If  the  brake  be  not  automatically  supported  at  its  top  it  will 


58  MACHINE  DESIGN 

lie  with  considerable  pressure,  due  to  its  own  weight,  on  the  bra.ke 
.surface  when  it  is  supposed  to  be  free  from  5t,  and  by  the  friction 
thereby  created  will  produce  a  heavy  drag  and  waste  of  power. 
A  spring  connection  fastened  to  an  overhead  beam  is  a  simple  way 
of  accomplishing  the  desired  result.  A  flat  supporting  strap  car- 
ried out  from  the  gear  guard,  having  some  degree  of  spring  in  it, 
is  a  neater  method  of  solving  the  problem.  The  spring  should 
be  just  strong  enough  to  counterbalance  the  weight  of  the  strap 
and  yet  not  resist  to  an  appreciable  degree  the  force  applied  to 
throw  the  brake  on. 

GENERAL  DRAWING. 

The  last  step  in  the  process  of  design  of  a  machine  is  the 
making  of  the  assembled  or  general  drawing.  This  should  be 
built  up  piece  by  piece  from  th'e  detail  drawings,  thereby  serving 
as  a  last  check  on  the  parts  going  together.  This  drawing  may 
be  a  cross  section  or  an  outside  view.  In  any  case  it  is  not  wise 
to  try  to  show  too  much  of  the  inside  construction  by  dotted  lines, 
for  if  this  be  attempted,  the  drawing  soon  loses  its  character  of 
clearness,  and  becomes  practically  useless.  A  general  drawing 
should  clearly  hint  at,  but  not  specify,  detailed  design.  It  is 
just  as  valuable  a  part  of  the  design  as  the  detail  drawing,  but 
it  cannot  be  made  to  answer  for  both  with  any  degree  of  success. 
A  good  general  drawing  has  plenty  of  views,  and  an  abundance  of 
;^oss  sections,  but  few  dotted  lines. 

The  general  drawing  of  the  machine  under  consideration  is 
left  for  the  student  to  work  up  from  the  complete  details  shown. 
It  would  look  something  like  the  preliminary  layout  of  Fig.  10,  if 
the  same  were  carefully  carried  out  to  finished  form.  A  plain  out- 
side  view  would  probably  be  more  satisfactory  in  this  case  than  a 
cross  section,  as  the  latter  would  show  little  more  of  value  than  the 
former.  The  functions  which  the  general  drawing  may  serve  are 
many  and  varied.  Its  principal  usefulness  is,  perhaps,  in  showing 
to  the  workman  how  the  various  parts  go  together,  enabling  him  to 
sort  out  readily  the  finished  detail  parts  and  assemble  them,  finally 
producing  the  complete  structure.  Otherwise  the  making  of  a 
machine,  even  with  the  parts  all  at  hand,  would  be  like  the  putting 
together  of  the  many  parts  of  an  intricate  puzzle,  and  much  time 


MACHINE  DESIGN  59 

would  be  wasted  in  trying  to  make  the  several  parts  fit,  with  per- 
haps never  complete  success  in  giving  each  its  absolutely  correct 
location.  • 

The  general  drawing  also  gives  valuable  information  as  to  the 
total  space  occupied  by  the  completed  machine,  enabling  its  loca- 
tion in  a  crowded  manufacturing  plant  to  be  planned  for,  its  con- 
nection to  the  main  driving  element  arranged,  and  its  convenience 
of  operation  studied. 

In  some  classes  of  work  it  is  a  convenient  practice  to  letter 
each  part  on  the  general  drawing,  and  to  note  the  same  letters  on 
the  specification  or  order  sheet,  thus  enabling  the  whole  machine 
to  be  ordered  from  the  general  drawings.  This  is  a  very  excel- 
lent service  performed  by  the  general  drawing  in  certain  lines  of 
work,  but  for  such  a  purpose  the  drawing  is  quite  inapplicable 
in  others. 

Merely  as  a  basis  for  judgment  of  design,  the  general  drawing 
fulfils  an  important  function  in  any  class  of  work,  for  it  approaches 
the  nearest  possible  to  the  actual  appearance  that  the  machine  will 
have  when  finished.  A  good  general  drawing  is,  for  critical  pur- 
poses, of  as  much  value  to  the  expert  eye  of  the  mechanical 
engineer  as  the  elaborate  and  colored  sketch  .of  the  architect  is  to 
the  house  builder  or  landscape  designer. 

From  the  above  it  is  readily  understood  that  the  general 
drawing,  although  a  mere  putting  together  of  parts  in  illustration, 
is  yet  of  great  assistance  in  producing  finished  and  exact  machine 
design. 

GENERAL  COMMENTS  ON  PRECEDING  PROBLEM. 

After  following  through  the  detail  of  work  as  given  in  the 
preceding  pages,  it  is  worth  while  to  stop  for  a  moment  and 
take  a  brief  survey  or  review  of  the  subject  as  illustrated  therein. 

If  the  text  be  carefully  studied  it  will  be  seen  that  in  every 
part  to  be  designed  the  same  routine  method  has  been  followed, 
regardless  of  the  final  outcome.  In  some  cases  it  may  seem  a 
roundabout  procedure  to  follow  a  train  of  thought  that  finally 
ends  in  a  design  apparently  based  on  purely  practical  judgment, 
the  theory  having  had  but  very  little  if  any  influence.  The  ques- 
tion at  once  arises — Why  not  use  the  empirical  rule  or  formula  in 


60  MACHINE  DESIGN 

the  first  place  ?  Why  not  make  a  good  guess  at  once  ?  Why  not 
save  all  the  time  and  energy  devoted  to  a  careful  analysis  and 
theory,  if  we  are  finally  to  throw  them  away  and  not  base  our 
design  on  them  ? 

The  principle  to  be  noted  in  this  connection  is,  that  it  is  just 
as  fatal  to  good  design  to  rely  upon  bare  experience  and  upon 
judgment  alone,  as  it  is  to  construct  solely  according  to  what  pure 
theory  tells  us.  There  are  many  things  in  the  operation  of 
machinery  that  are  totally  inexplicable  from  the  purely  practical 
point  of  view,  and  will  forever  remain  so  until  we  analyze  them 
and  theorize  on  them.  Many  good  things  in  machinery  have 
been  the  result  of  what  might  be  called  "  reversed "  machine 
design.  When  a  new  machine  is  started,  it  frequently,  or  we 
might  almost  say  always,  fails  to  do  its  work  just  as  it  is  expected 
to  do  it.  This  is  because  some  little  point  of  design  is  bad,  owing 
to  the  inability  of  drawings,  however  good  they  may  be,  to  show 
all  that  the  machine  itself  in  bodily  form  and  in  motion  shows. 

Now,  if  our  analysis  and  theory  have  been  good  in  the 
designing  process,  it  is  almost  sure  that  we  can  very  readily 
analyze  and  theorize  on  the  trouble  that  exists  when  the  machine 
is  finished,  can  detect  the  weakness,  and  can  correct  it  with  com- 
paratively small  change  in  the  general  design.  This  is  "  reversed  " 
machine  design. 

If,  on  the  contrary,  we  have  based  our  design  purely  on  guess- 
work, allowing  our  fancy  full  and  free  play  to  work  out  the  details 
without  further  basis,  we  may  consider  ourselves  lucky  if  the 
machine  runs  at  all.  This,  however,  is  not  the  worst  of  the 
situation.  If  the  machine  does  actually  operate,  even  as  well  as 
it  might  reasonably  be  expected  to,  but  still  has  the  usual  diffi- 
culty of  some  little  kink  or  hitch  that  was  not  expected,  then,  as  a 
result  of  the  method  upon  wilich  the  whole  thing  has  been  con- 
structed, we  have  no  definite  plan  of  action  to  proceed  upon.  We 
must  try  first  this,  then  that  scheme  to  obviate  the  trouble.  We 
may  be  fortunate  enough  to  "  strike  it "  the  first  time  ;  we  may 
never  strike  it.  It  is  doubtful  if  the  machine  ever  can  be  made  to 
work  at  highest  efficiency  ;  and  if  fairly  good  results  be  finally 
obtained  we  never  know  the  reason  why,  and  have  nothing  on 
which  to  base  any  future  action  or  design. 


MACHINE  DESIGN  61 

This  haphazard  process  is  not  machine  design  at  all,  either 
in  name  or  in  result. 

As  has  previously  been  stated  in  these  pages,  there_is  no 
such  thing  as  too  much  analysis  or  theory  in  the  designing  of 
machinery.  Even  if  we  carefully  analyze,  theorize  with  rigorous 
exactness,  and  then  practically  modify  our  construction  to  such  a 
point  that  the  original  theoretical  shape  is  almost  or  entirely  lost, 
the  apparently  roundabout  process  is  not  in  vain,  for  we  are  in  per- 
fect control  of  our  design.  We  know  exactly  what  it  has  to  take  in 
the  way  of  forces,  blows  and  vibrations.  "We  know  what  its  ideal 
shape  should  be.  We  know  where  we  can  practically  modify  its 
form  without  weakening  it  excessively  or  adding  excess  of  material. 
In  other  words  we  know  all  about  it,  and  therefore  know  exactly 
what  we  can  do  with  it  ;  and  whether  it  follows  in  its  shape  the 
outline  that  pure  theory  gives  it  or  some  other  outline,  it  is  never- 
theless well  designed. 

"Reversed"  machine  design,  as  described  above,  based  on 

O      '  ' 

observation  and  experiment  with  regard  to  machines  already  in 
operation,  is  just  as  impossible  without  exact  analysis  and  theory 
as  is  original  design  based  merely  on  mechanical  ideas  in  the 
abstract.  The  method  once  learned  and  made  a  habit  of  mind 
will  produce  results  with  equal  facility  in  either  case,  and  results 
are  what  the  mechanical  world  is  seeking;. 

o 

Another  point  worth  noting  in  the  progress  of  the  problem 
as  given  is  the  absolute  necessity  of  possessing  some  knowlege  of 
Mechanics.  The  more  of  this  subject  the  designer  can  have  at 
his  finger  ends,  the  more  ready  and  successful  will  he  be  in  all 
problems  of  Machine  Design.  However,  the  principles  of  forces 
and  moments  clearly  understood,  and  the  application  of  the  same 
in  the  all-important  subject,  "Strength  of  Beams,"  constitute  a 
fund  of  information  that  will  give  a  splendid  start  and  a  good 
working  basis  for  simple  designs.  It  should  always  be  remem- 
bered that  a  complicated  design  is  little  more  than  a  combination 
of  simple  designs,  and  if  one  has  the  ability  to  dissect  and  analyze 
what  seems  at  first  like  a  bewildering  maze  of  parts,  complication 
is  speedily  changed  to  simplicity. 

Common  sense  goes  a  long  way  in  good  designing.  There  is 
nothing  mysterious  about  the  process  If  the  beginner  will  only 


62  MACHINE  DESIGN 

avoid  doing  things  that  are  foolish  and  ridiculous  on  their  very 
face,  if  he  will  exercise  the  same  judgment  that  he  uses  in  the 
daily  affairs  of  his  life  and  will  mix  in  something  of  mechanics  and 
mechanical  method,  he  will  be  on  the  direct  road  to  success  in  the 
art. 

Good  drawing  is  an  essential  element  of  good  design,  and  it 
is  especially  urged  that  the  sketches  and  drawings  as  reproduced 
in  the  preceding  text  be  studied  with  this  in  mind.  By  a  good 
drawing  is  meant  not  a  showy  piece  of  work,  finely  shaded  or 
artistically  lettered,  but  an  exact  layout,  definite  and  measurable, 
correctly  dimensioned  if  in  detail,  and  meaning  exactly  what  it 
says.  Machine  design  is  an  exact  science,  and  the  designer  can- 
not shirk  responsibility  by  permitting  his  work  to  be  shiftless  and 
loose.  If  he  cannot  delineate  clearly  and  in  definite  form  what  he 
determines  in  his  mind  the  structure  should  be,  then  it  is  purely 
good  luck  if  he  achieves  success,  and  it  may  safely  be  asserted  that 
the  success  is  due  to  some  subsequent  care  and  finished  design 
added  to  his  feeble  effort,  rather  than  to  any  expertness  of  his  own. 
Such  success  is  of  a  very  doubtful  nature,  and  if  not  bordering  on 
financial  loss  it  is  at  least  secured  only  at  a  low  working  efficiency. 

As  examples  of  good  drawings  the  plates  shown  are  not 
claimed  to  be  anything  extraordinary,  but  it  will  be  noted  that  they 
are  clean-cut  and  definite,  and  that  even  the  sketches  are  unmis- 
takable as  to  that  which  they  are  intended  to  illustrate.  The 
information  as  to  the  design  is  all  there;  nothing  is  left  to  the 
imagination. 

Classification  of  Machinery.  It  is  intended  to  be  made  clear 
in  all  that  has  preceded,  that  the  same  method  of  attack  and  pro- 
cedure may  be  applied  to  the  designing  of  machinery,  whatever 
may  be  tLe  class  or  kind.  This  is  a  fundamental  principle. 
When  it  is  logically  carried  cut,  however,  it  produces  very  differ- 
ent results,  as  is  evidenced  by  the  characteristics  of  style  peculiar 
to  each  of  the  classes  of  machinery  to  one  or  another  of  which 
all  machines  belong. 

For  example,  an  engine  lathe  has  a  style  similar  to  a  drill 
press,  or  a  boring  mill,  or  a  screw  machine,  or  a  milling  machine. 
It  is  very  different,  however,  from  the  style  of  a  steam  engine,  or 
a  pump,  or  an  air  compressor,  or  a  locomotive;  it  is  still  more  dif- 


MACHINE  DESIGN  63 


ferent  from  the  style  of  a  rolling  mill,  or  a  link  belt  conveyor,  or 
a  coal  crusher,  or  a  stamp  mill. 

These  classes  of  machinery  are  so  distinctly  marked—that  the 
novice  is  easily  able  to  perceive  that  there  is  some  controlling 
influence  in  each  which  marks  its  peculiar  style.  He  should  at 
the  same  time  see  that  the  very  analysis  that  has  been  so  strongly 
insisted  upon  in  these  pages  is  the  direct  cause  of  the  marked 
characteristic  in  design.  Each  class  of  machinery  must  satisfy 
certain  exacting  conditions  different  from  those  of  any  other,  and 
it  is  the  careful  study  of  these  conditions,  as  fundamentally 
enforced,  which  leads  to  the  strictly  logical  design. 

A  few  of  the  most  common  classes  are  enumerated  below, 
and  their  prominent  features  noted.  It  is  hoped  that  a  study  of 
them  will  familiarize  the  student  in  a  general  way  with  the 
requirements  of  each,  and  serve  as  a  guide  to  a  more  comprehen- 
sive study  of  their  detail  design  than  is  possible  in  these  pages. 

Machine  Tools.  Examples: — lathe,  planer,  milling  machine, 
drill  press,  screw  machine,  boring  mill,  grinding  machine,  etc.,  etc. 

The  machines  of  this  class  are  all  utilized  for  the  finishing  of 
metal  surfaces.  They  are  really  at  the  root  of  the  production  of 
machinery  of  all  other  classes.  Accuracy  is  their  prime  character- 
istic— accuracy  of  construction,  accuracy  of  operation,  accuracy  of 
adjustment.  Any  inaccuracy  that  exists  primarily  in  a  machine 
tool  is  reproduced  in  every  piece  upon  which  it  produces  a  finished 
surface  ;  and  since  the  mere  act  of  finishing  a  surface  upon  any- 
thing implies  that  a  rough  and  inaccurate  surface  will  not  answer, 
the  tool  then  fails  of  its  purpose  if  it  cannot  produce  a  true  sur- 
face: it  does  not  accomplish  that  for  which  it  was  designed. 

The  effect  that  this  element  of  accuracy  has  upon  the  design 
of  a  machine  tool  is  to  require  long  bearings,  convenient  and  exact 
methods  of  adjustment,  stiffness,  excess  of  material  to  absorb 
vibration,  special  shapes  to  facilitate  application  of  jigs,  fixtures, 
and  exact  manufacturing  devices  insuring  interchangeabilit^  of 
parts,  dust  guards,  and  automatic  lubrication. 

Machine  tools  are  essentially  machines  of  maximum  output, 
and  depend  for  their  success,  not  only  upon  their  accuracy  as 
noted,  but  also  upon  their  ability  to  do  the  greatest  amount  of  work 
per  square  foot  of  space  occupied,  with  the  least  amount  of  manual 


64  MACHINE  DESIGN 

labor  and  attention  on  the  part  of  the  operator.  This  is  especially 
true  of  automatic  machinery,  which  perhaps  might  be  classed  by 
itself  in  this  respect,  but  which  is  nevertheless  included  under  the 
broad  term  of  a  machine  for  producing  finished  surfaces,  being 
merely  the  highest  and  most  refined  form  of  same.  For  machines 
of  this  class  the  designer  has  to  study  every  detail  with,  the  most 
minute  attention,  packing  away  the  operating  parts  into  the 
smallest  space  and  yet  providing  ready  means  for  access,  removal, 
and  repair.  Clearances  that  would  be  too  little  for  other  kinds  of 
machinery  are  permitted  and  provided  for;  material  of  high  grade, 
strength,  and  wearing  quality,  though  expensive  in  first  cost,  and 
requiring  the  most  expert  skill  to  finish  and  to  fit  into  place,  must 
be  used  in  order  to  keep  the  machine  compact  and  yet  of  large 
capacity,  to  make  it  reasonably  light  in  weight  and  yet  amply 
strong. 

Another  point  which  has  a  great  influence  on  the  design  of  a 
machine  tool  is  that  we  can  never  tell  in  advance  just  what  it  will 
have  to  stand  in  wrork,  for  the  variation  in  the  material  that  it  fin- 
ishes, the  uncertain  skill  of  the  operator  who  runs  it,  the  crowding 
to  its  limit  of  capacity  and  even  beyond  in  times'of  press  of  business, 
and  the  many  other  stresses  that  may  suddenly  and  without  warn- 
ing be  thrown  upon  it,  must  all  be  thought  of  and  provided  for. 

The  points  above  mentioned  are  but  a  few  of  those  which  the 
designer  of  machine  tools  has  to  meet,  and  are  presented  merely 
as  illustrations  to  show  the  special  skill  required  in  this  class  of 
machinery.  It  is  readily  seen  that  while  the  machine  tool 
designer  has  great  latitude  in  choice  of  material  and  in  expendi- 
ture of  money  for  refinement  of  structure — perhaps  greater  lati- 
tude than  in  any  other  class,  yet  he  is  held  down  as  in  no  other 
to  the  final  productive  results,  a  small  percentage  of  failure  entirely 
throwing  out  the  machine  as  a  marketable  product. 

The  style  and  external  appearance  of  machine  tools  have  a 
character  of  their  own  resulting  from  this  extreme  detailed  care  in 
design.  Corners  and  fillets  are  carefully  rounded;  surfaces  and 
intersections  are  definitely  made;  in  short,-the  mechanical  beauty 
of  a  machine  tool  is  seen  only  from  a  near  view  and  close  inspec- 
tion, and  it  is  to  this  end  that  the  design  is  constantly  directed 
Appearance  is  a  large  factor  in  the  sale  of  a  fine  tool,  and  the 


MACHINE  DESIGN  65 

prestige  of  the  American  trade  abroad  in  this  respect  is  very 
noticeable. 

Motive=Power  flachinery.  Examples :— Steam  engine,  gas 
engine,  air  compressor,  steam  pump,  hydraulic  machinery,  etc.,  etc. 

The  element  of  heat  enters  into  the  design  of  all  machinery 
in  this  class.  The  natural  agents,  air,  gas,  and  water,  in  their 
various  forms,  are  taken  into  the  machine  in  the  most  efficient 
form  in  which  it  is  possible  to  obtain  them,  are  robbed  of  their 
energy  to  provide  power,  and  are  discharged  in  a  form  as  weak 
and  inert  as  the  efficiency  of  the  machine  will  determine. 

In  contrast  to  the  class  of  machinery  just  studied,  it  should 
be  noted  that  these  machines  do  not  produce  any  material  thing; 
that  is,  they  do  not  produce  finished  surfaces  on  metals,  make 
screws  or  bolts,  bore  holes  in  castings,  or  turn  line  shafting. 
They  merely  take  the  energy  of  the  natural  agent,  which  is  not  in 
a  form  available  for  use,  and  transform  it  into  motive  power  for 
general  use. 

Hence  the  element  of  accuracy  as  entering  into  the  design  of 
these  machines  is  necessary  only  for  their  own  efficient  operation, 
and  not  for  the  quality  of  the  thing  which  they  produce,  as  in  the 
6ase  of  machine  tools.  For  example,  the  power  furnished  by  one 
steam  engine  to  drive  a  line  shaft  is  as  good  as  that  of  another  as 
far  as  the  rotating  of  the  shaft  is  concerned,  provided,  of  course, 
that  both  are  equipped  with  the  same  quality  of  governing  mechan- 
ism. The  fact  that  one  of  the  engines  has  a  good  adjusting  device 
on  the  main  bearing  while  the  other  has  not  is  of  no  consequence 
from  the  standpoint  of  tihe  line  shaft,,  but  it  is,  of  course,  of  con- 
sequence  respecting  the  efficient  operation  of  the  engines. 

The  design  of  steam  engines  and  similar  machines  is  of  a 
rough  nature  compared  with  that  of  machine  tools,  as  far  as  the 
detail  of  surface  is  concerned.  General  accuracy  is  nevertheless 
essential  for  the  machine's  own  sake,  but  while  in  the  machine 
tool  wTe  deal  with  thousandths  of  an  inch,  in  the  steam  engine 

o 

hundredths  of  an  inch  indicates  fine  work. 

These  machines  are  subject  to  extremes  of  temperature  that 
have  to  be  provided  for  in  the  design  and  arrangement  of  the  parts. 
Being  prime  movers,  controlling  the  operation  of  many  machines, 
they  must  be  certain  to  run  during  their  period  of  work;  hence 


66  MACHINE  DESIGN 

design  and  'adjustment  must  be  positive,  and  when  the  latter  can- 
not be  made  while  running,  it  must  be  quickly  and  definitely  accom- 
plished when  a  stop  is  made.  Simplicity  of  construction  is  essential, 
facilitating  cheap  and  quick  repairs.  The  design  should  be  such 
that  constant  attention  while  running  is  avoided,  the  usual  atten- 
tion of  the  engineer  being  a  safeguard  rather  than  an  implied  fac- 
tor of  the  original  design.  General  rigidity  and  stiffness  are 
important,  also  good  balancing  of  the  moving  parts,  and  weight  for 
absorption  of  vibration ;  otherwise  under  the  constant  daily  run  the 
machines  will  tear  to  pieces  not  only  themselves  but  their  founda- 
tions. 

As  far  as  external  appearance  goes  in  this  and  subsequent 
classes  to  be  mentioned  we  are  on  a  very  different  basis  from  that 
of  machine  tools.  General  mechanical  symmetry  of  form  is  aimed 
at  in  the  design,  and  the  several  smaller  parts  depend  for  their  out- 
line (aside  from  considerations  of  strength,  which  are,  of  course, 
always  in  order)  upon  the  harmonious  relation  which  they  bear  to 
the  main  and  fundamental  elements  of  the  machine.  Such 
machinery  as  air  compressors,  steam  engines,  pumps,  and  the  like 
are  viewed  as  a  whole,  and  criticised,  not  detail  by  detail,  as  is  the 
machine  tool,  but  as  to  general  effect  of  outline  observed  from 
some  distance.  To  convey  the  desired  effect  to  the  eye  the  design 
must  be  bold  and  massive,  connections  simple  and  direct,  and  the 
smaller  parts  must  not  be  so  dwarfed  in  size  as  to  appear  like  deli- 
cate ornaments  instead  of  integral  parts  of  the  machine.  The  lines 
of  connected  parts  must  be  continuous  from  one  part  to  the  other; 
and  when  interrupted  by  flanges.,  bosses,  or  lugs,  the  latter,  which 
are  merely  incidental  to  the  former  must  not  be  allowed  to  obscure 
wholly  the  main  lines  of  the  fundamental  pieces. 

It  is  attention  to  such  points  as  thes^  that  marks  the  difference 
between  well-designed  motive-powrer  iKcuhinery  and  that  of  the 
opposite  character.  Even  though  the  little  details  of  fillets  and 
corners  and  surfaces  may  have  their  effect  from  a  close  point  of 
view,  the  design  will  stand  or  fall  in  excellence  on  its  bolder 
features,  as  noted  above. 

Structural  Machinery.  Examples: — Hoists,  cranes, elevators, 
transfer  tables,  locomotives,  cars,  conveyors,  cable-ways,  etc.,  etc. 

In  the  two  preceding  classes  that  have  been  noted,  cast  iron 


MACHINE  DESIGN  67 

in  the  form  of  foundry  castings  enters  as  the  principal  material. 
Steel  is  utilized  for  shafts,  studs,  pins,  and  keys.  Also  special 
forgings,  malleable  iron  and  steel  castings  enter  as  factors  in  the 
production  of  the  machinery  discussed.  Foundry  castings,  how- 
ever, compose  the  great  body  of  the  material  used,  and  the  chief 
problems  involved  are  those  of  the  expert  moulding  of  cast  iron, 
and  the  handling  and  finishing  of  the  same.  For  the  operating 
parts,  steel  of  fine  grade  is  used  in  highly  finished  form,  expens- 
ive because  of  its  fineness,  and  yet  a  necessity  to  the  extent  it  is 
used.  Brass  and  bronze  are  used  in  the  same  way,  generally  in 
connection  with  the  bearings  for  the  shafts. 

Structural  machinery,  on  the  contrary,  uses  steel  as  the  basis 
of  its  construction.  The  fundamental  structure  is  built  up  of 
plates,  channels,  beams,  and  angles;  castings,  though  numerous, 
are  relatively  small,  being  riveted  or  bolted  to  the  main  structure 
and  controlled  In  their  design  by  its  requirements. 

Steel  is  used  in  this  manner  partly  because  the  exclusive  use 
of  castings  is  prohibited  on  account  of  the  excessive  weight,  and 
therefore  expense,  and  partly  because  castings  could  not  be  made 
which  wrould  possess  the  necessary  toughness  and  strength.  In 
many  cases  the  size  of  the  machinery  is  such  that  castings,  even 
if  they  could  be  made,  would  not  support  their  owrn  weight. 
Moreover,  machinery  of  this  class  is  subjected  to  rough  service, 
and  yet  must  be  practically  infallible  under  all  conditions,  neither 
being  uncertain  in  operation  at  critical  moments  nor  entirely  fail- 
ing under  an  extraordinary  load. 

The  design  of  structural  machinery -is  tied  up  to  con- 
ditions existing  largely  outside  of  the  locality  in  which  the  ma- 
chinery is  built.  The  steel  plates  and  structural  shapes  required, 
being  products  of  the  rolling  mill,  have  to  conform  to  the  latter's 
standards.  The  rivets,  bolts  and  other  fastenings  have  to  be  in 
accordance  with  the  established  practice  of  the  structural  iron 
worker,  in  order  to  permit  punching,  shearing  and  bending  ma- 
chinery of  regular  form  to  be  utilized.  Shipment  on  standard 
railway  cars  has  to  be  considered,  the  design  often  requiring  to  be 
modified  to  permit  this  and  nevertheless  insure  positive  and 
accurate  assembling  in  the  field. 

Steel  castings,  both  large  and  small,  find  ready  application  in 


68  MACHINE  DESIGN 

this  class  of  work;  also  steel  forgings,  requiring  to  be  worked 
under  a  heavy  hammer  and  in  many  cases  by  specially  devised 
processes. 

In  structural  design  less  of  the  actual  process  of  manufacture 
is  under  the  eye  of  the  designer  than  in  the  former  classas  of 
machinery  which  have  been  considered,  and  hence  more  allowance 
has  to  be  made  for  things  not  coming  exactly  right  to  the  fraction 
of  an  inch.  It  would  be  bad  design  to  plan  any  structural  piece 
of  work  with  the  same  closeness  of  detail  permitted,  and  in  fact 
required,  in  the  case  of  machine  tools,  or  even  in  the  case  of  motive- 
power  machinery.  In  planning  structural  work  the  idea  must  be 
carried  out,  of  certainty  of  operation  in  spite  of  roughness  of  detail 
and  variations  of  construction.  This  does  not  necessarily  imply 
inaccuracy,  or  shiftless,  loosely  constructed  machinery;  on  the  con- 
trary, quite  the  reverse.  The  locomotive,  for  example,  is  one  of 
the  most  refined  pieces  of  mechanism  that  exists  today;  and  yet 
the  methods  applied  to  the  construction  of  machine  tools  would 
prove  a  failure  on  the  locomotive.  The  design  of  a  car  axle  box 
has  to  be  just  right  else  it  will  heat  and  destroy  itself;  the  same  is 
true  of  the  spindle  of  a  fine  engine  lathe;  and  yet  how  rough  the 
former  is  compared  with  the  latter,  and  how  unsuited  either  would 
be  for  use  on  the  service  of  the  other. 

As  a  general  rule  structural  machinery  can  be  more  closely 
proportioned  to  theoretically  calculated  size  than  can  the  preceding 
types.  The  rolled  material  of  which  it  is  made  is  of  a  uniform 
and  homogeneous  nature  owing  to  its  process  of  manufacture, 
hence  its  every  fibre  may  be  counted  on  to  sustain  its  share  of  the 
total  load  imposed  upon  it.  This  is  in  sharp  contrast  to  the  case 
of  cast  iron,  which  is  of  such  a  porous  and  irregular  structure  that 
we  have  to  use"  a  large  factor  of  safety  to  cover  this  inherent 
defect. 

Steel  castings  of  both  small  and  large  size  (which  are  quite 
apt  to  be  utilized  in  this  class  of  machinery  for  parts  that  can  with 
difficulty  be  made  out  of  rolled  material),  if  properly  designed  of 
uniform  thickness,  with  all  corners  well  filleted  and  with  the 
channels  for  the  flow  of  the  molten  metal  direct  and  ample,  are 
nearly  as  reliable  as  rolled  steel.  In  parts  subject  to  excessive 
vibration,  shocks,  and  sudden  wrenchings,  as,  for  example,  the 


MACHINE  DESIGN  69 

side  frames  or  the  connecting  rod  of  a  locomotive,  the  forged  and 
hammered  material  is  practically  a  necessity.  This  is  especially 
the  case  when  the  possible  breakage  of  the  part  wo~ul6T  cause 
serious  consequences  involving  heavy  loss  of  life  and  property. 

From  the  several  points  of  view  as  above  considered,  it  can 
be  readily  appreciated  that,  while  structural  work  is  in  one  sense 
rough  and  unpolished,  yet  it  requires,  from  an  engineering  stand- 
point, quite  as  much  breadth  of  experience  and  judgment  as  any 
of  the  other  types.  The  fine-tool  designer,  least  of  all,  perhaps, 
requires  book  theory,  but  does  require  an  extended  machine-shop 
experience.  The  designer  of  motive-power  machinery  needs  pure 
physical  theory  and  shop  experience  of  a  large  and  broad  scope. 
The  structural  designer  is  least  of  all  concerned  with  refined  and 
minute  finishing  processes,  but  utilizes  his  theory  absolutely,  even 
though  roughly. 

Mill  and  Plant  Machinery.  Examples: — Rolling  mills, 
mining  machinery,  crushers,  stamps,  rock  drills,  coal  cutters,  the 
machinery  of  blast  furnaces  and  steel  mills,  tube  mills,  etc.,  etc. 

This  machinery  constitutes  a  class  which  in  the  roughness 
of  its  operation  exceeds  all  others.  Moreover,  it  is  machinery 
which  for  the  most  part  is  in  continuous  operation — 24  hours  per 
day  and  365  days  in  the  year.  Hence  refinement,  even  such  as 
might  be  permitted  in  the  preceding  class  of  Structural  Machinery, 
would  be  fatal  here.  The  conditions  that  surround  plant  machinery 
are  unfavorable  in  the  extreme  to  the  life  of  any  material  or  metal, 
and  it  is  not  possible  to  change  these  conditions  or  give  more 
than  partial  protection  to  the  operating  parts.  Hence  the  design 
of  such  machinery  must  proceed  primarily  on  the  assumption 
that  abuse  and  neglect,  grinding  away  of  surfaces,  chemical  eating 
away  of  metal,  flooding  of  parts  with  water  gritty  and  corrosive, 
subjection  to  sudden  bursts  of  flame  and  intense  heat,  etc.,  will 
in  a  relatively  short  time  totally  destroy,  perhaps,  the  entire 
structure. . 

In  view  of  the  continuous  nature  of  the  working  process, 
which  must  be  kept  up  in  spite  of  these  almost  insurmountable 
conditions,  the  problem  in  each  case  becomes  one  of  expediency; 
and  the  designs  and  arrangement  of  machinery  must  be  so  worked 
out  that  operation,  repair,  construction,  and  installation  can  all  go 


70  MACHINE  DESIGN 

on  simultaneously  without  stopping  the  continuous  process,  and 
with  but  a  small  degree  of  inconvenience  to  the  operation  of  the 
plant. 

This  problem,  difficult  though  it  may  seem,  can  be  worked 
out  successfully,  as  is  evidenced  by  the  great  number  of  plants  of 
the  continuous  character  operating  at  high  efficiency  throughout 
the  world.  The  engineering  and  designing  skill  required  to  ac- 
complish this,  is  perhaps  of  the  highest  degree  met  with  in  mod- 
ern practice,  for  in  it  is  involved  a  working  knowledge  of  the 
possibilities,  if  not  the  detailed  designs  of  machinery  included  in 
all  classes.  And  yet,  as  in  the  most  elementary  case  of  simple 
design  that  can  be  conceived,  the  result  is  accomplished  in  the 
same  way,  namely,  by  studying  the  conditions  (analysis),  devel- 
oping an  ideal  application  to  those  conditions  (theory),  and  then 
reducing  the  ideal  design  to  a  practical  basis  (modification), 

A  Few  Pointed  Suggestions  on  Original  Design.  Original 
design  deals  with  the  development  of  original  mechanical  ideas. 
The  prime  requisite  for  the  development  of  an  idea  is  to  under- 
stand thoroughly  the  idea  in  the  rough.  See  distinctly  the  mark 
aimed  at,  and  never  lose  sight  of  it.  If  a  method  of  reaching  it 
is  already  outlined,  understand  that  also  thoroughly  and  the  prin- 
ciples involved.  It  is  impossible  to  go  ahead  blindly  and  hope  to 
come  out  right.  No  good  machine  was  ever  built  that  does  not 
stand  for  hours  of  concentrated  thought  on  the  part  of  its  designer. 
Good  machines  never  happen,  they  always  grow. 

Just  as  soon  as  the  object  to  be  accomplished  is  clearly  under- 
stood, begin  to  produce  some  visible  work  on  the  problem.  Sketch 
something.  Get  some  ideas  on  paper.  Ideas  on  paper  suggest 
other  ideas.  If  the  problem,  for  example,  is  one  of  lathe 
design,  sketch  a  rectangle,  and  call  it  the  headstock;  another  rec- 
tangle, and  call  it  the  footstock;  a  couple  of  scratches  for  the 
centers;  some  steps  for  the  cone  pulley;  three  or  four  lines  for  the 
bed;  and  as  many  more  for  the  supports.  There  is  now  something 
on  paper  to  look  at;  the  design  is  begun. 

It  is  much  better  to  stare  at  this  sketch,  than  into  blank 
space  trying  to  imagine  the  finished  design.  No  matter  how  rough 
the  sketch  may  be,  a  short  study  of  it  will  develop  some  limiting 
conditions  that  before  were  not  apparent.  Guess  at  a  few  rough 


MACHINE  DESIGN  71 

dimensions;  put  them  on  the  sketch;  develop  another  view — a  plan 
or  a  side  elevation — all  still  in  the  roughest  style,  without  any 
regard  to  finished  detail.  Information  will  be  growing  all  the 
while,  and  the  problem  will  be  opening  up.  At  this  "stage  it  is 
probable  that  the  sketch  can  easily  be  seen  to  be  wrong  in  many 
respects.  Perhaps  the  arrangement  will  not  do  at  all. 

This  is  a  good  sign.  It  shows  that  the  design  is  progressing. 
It  is  a  valuable  thing  to  know  that  certain  plans  cannot  be  fol- 
lowed. Do  not  rub  out  part  of  the  sketch  already  made  and  try 
to  remedy  it.  Begin  again.  Make  another  sketch.  Sketch  paper 
is  cheap.  By  and  by  it  may  prove  to  be  very  desirable  to  have 
that  first  rough  outline  available  for  comparison ;  or  it  may  be  that 
some  of  its  ideas  can  be  applied  on  other  sketches.  The  second 
sketch  may  "show  up"  little  or  no  better  than  the  first.  Make 
another,  and  another,  and  another,  until  the  subject  is  thoroughly 
digested.  It  is  wonderful  how  helpful  it  is  to  have  some  marks 
0:1  paper  relative  to  a  design,  even  though  they  be  of  the  utmost 
crudeness.  They  save  imaginative  power  tremendously;  and,  even 
with  them,  all  available  powers  of  imagination  will  be  needed 
before  the  design  is  perfected. 

A  careful  comparison  of  one's  sketches,  rejecting  here,  and 
approving  there,  will,  little  by  little,  bring  about  a  definite  opinion, 
and  the  scale  drawing  can  be  begun. 

As  in  the  case  of  the  first  sketch,  so  iii  the  case  of  the  first 
scale  drawing,  get  some  lines  on  paper  as  quickly  as  possible. 
Draw  something,  even  if  it  is  nothing  more  than  a  straight  hori- 
zontal line.  Do  not  stare  at  blank  paper  for  an  hour  trying  to 
imagine  how  the  tenth  or  eleventh  line  is  going  to  be  drawn  in 
relation  to  the  first  line.  Do  not  worry  about  the  later  lines 
until  it  is  time  for  them  Draw  the  first  line  at  once;  and,  when 
the  second  line  is  drawn,  if  the  first  line  proves  to  be  wrong, 
make  it  .right.  As  in  the  rough  sketch,  that  first  horizontal  line 
is  an  immense  relief  from  the  great  waste  of  blank  paper  of  a 
fresh  sheet.  It  is  something  to  look  at.  It  is  the  beginning  of  a 
detailed  design.  If  it  happens  not  to  be  the  absolutely  correct 
foundation  to  build  upon,  it  at  least  is  something  to  tear  down. 
The  main  purpose  of  these  preliminary  drawings  is  to  keep  the 
mind  active  on  the  problem*,  and  advance  toward  the  final  accoin- 


72  MACHINE  DESIGN 

plishment  of  the  design  is  often  made  quite  as  rapidly  by  discover- 
ing what  to  tear  down  as  by  consistently  building  up. 

When  a  detail  draftsman  who  has  been  used  to  having  all  his 
work  laid  out  for  him  by  an  expert  designer  attempts  to  take  up 
original  work  for  himself,  he  encounters  the  drawing  of  that  first 
line  in  a  way  he  never  did  before.  He  is  apt  to  ^orry  for  some 
time  over  the  possible  or  impossible  results  of  drawing  that  first 
line.  If  he  continue  this,  he  will  be  sure  to  fail.  The  second  line 
is  much  easier  to  draw  than  the  first,  and  the  third  than  the  second; 
and  the  next  hundred  will  follow  on  in  comparatively  smooth 
sequence,  all  because  of  bold  action  on  the  first  few  lines. 

And  yet,  just  as  the  design  appears  to'be  progressing  smoothly, 
and  the  advanced  progress  of  the  drawing  seems  cause  for  congratu- 
lation, careful  consideration  may  disclose  a  "snag"  not  previously 
known  to  exist  in  the  problem.  Further  study  pursued  along 
the  line  of  this  new  discovery  may  showr  that  the  whole  layout 
thus  far  lias  been  radically  wrong,  and  that  a  fresh  start  will  have 
to  be  made.  At  such  a  time  the  young  designer  is  apt  to  feel 
that  his  labor  has  all  been  thrown  away,,  and  he  becomes  discour- 
aged. There  is,  however,  no  cause  for  discouragement.  Machine 
Design  might  almost  be  defined  to  be  the  "successful  elimination 
of  snags."  It  takes  some  ability  to  discover  an  obstacle  of  this 
sort;  to  know  a  "snag"  when  an  opportunity  to  see  it  is  given. 
It  takes  a  good  designer  to  eliminate  such  a  difficulty  after  it  has 
been  found.  If  there  were  no  "snags"  it  would  not  require  great 
ability  to  design  machines.  Many  machines  fail  because  in  them 
there  are  a  lot  of  undiscovered  "snags."  Others  fail  because  the 
"snags,"  although  discovered,  were  not  eliminated  by  careful  design. 

Do  not  be  afraid  to  make  a  lot  of  "first"  drawings.  It  is 
just  as  important  to  digest  the  design  thoroughly  by  means  of 
scale  drawings,  as  it  was  to  digest  it  originally  by  means  of 
the  rough  sketches.  An  attempt  to  make  the  first  drawing  of  an 
original  design  absolutely  right  would,  it  is  safe  to  say,  produce  a 
poor  design,  one  that  could  be  much  improved  by  further  trial. 
Let  the  drawings  multiply,  one  after  another,  until  the  final  one 
is  reached,  in  which  the  perfection  of  detail  will  eliminate  all  the 
bad  points  of  the  preceding  drafts  and  incorporate  good  ones  of  its 
own  based  on  the  study  of  the  others. 


MACHINE  DESIGN  73 


And  yet  it  is  often  true  that  the  first  design  laid  out,  even 
after  many  others  have  been  developed,  may  -be  found  to  possess 
features  that  render  a  return  to  it  desirable.  This  is_why  it  is 
always  better  to  produce  a  collection  of  designs  than  to  attempt 
to  rub  out  and  work  over  the  first  one.  The  best  designers  usually 
have  a  great  number  of  sketches  showing  how  to  accomplish  a 
single  result.  Likewise,  they  also  have  a  series  of  layouts  to  scale, 
showing  in  detailed  form  the  development  of  their  various  ideas. 
This  is  because,  without  a  careful  consideration  of  many  methods, 
they  themselves  feel  incompetent  to  judge  of  the  best  design  pos- 
sible for  accomplishing  a  given  result. 

Sketches  and  original  designs  should  always  be  dated  and 
signed.  Different  designers  may  be  working  on  the  same  prob- 
lem, and  priority  of  design  will  never  be  allowed  except  upon 
signed  and  witnessed  papers.  It  is  embarrassing  to  find,  after 
months  and  perhaps  years  have  passed  since  an  original  drawing: 
was  made,  that  one's  rights  have  been  preempted  merely  because 
there  was  no  date  or  signature  to  define  them. 

In  redesigning  or  modifying  an  existing  machine,  never  make 
a  change  merely,  for  the  sake  of  doing  so.  Give  the  good  points  of 
the  machine  a  chance,  and  devote  attention  in  the  new  design  to 
correcting  the  bad  points.  It  is  in  bad  taste,  if  it  be  not  actually 
childish,  to  "look  wise  and  suggest  a  change"  in  details  which 
happen  to  have  been  designed  by  another  party,  but  which,  never- 
theless, are  by  common  engineering  judgment  pronounced  good 
for  the  special  work  intended.  This  element  of  unfair  and  selfish 
criticism  has  more  than  a  moral  bearing.  When  it  is  carried  into 
the  superintendence  of  designing  work,  it  extinguishes  the  person- 
ality of  the  subordinate  draftsman;  his  efficiency  as  an  original 
thinker  is  lowered;  and  narrow  designs  are  produced. 

"  The  best  way  for  a  subordinate  to  dispose  of  what 
appears  to  be  a  poor  suggestion  from  a  superior,  is  to  work  it  out 
to  the  best  degree  possible."  If  it  turns  out  to  be  good  the 
credit  of  working  it  out  belongs  to  the  man  who  did  it.  If  it  is 
actually  bad,  a  careful  working  out  will  usually  develop  the  fact 
beyond  dispute,  and  save  unprofitable  argument.  For  the  success 
or  failure  of  a  machine  there  is  only  one  argument  better  than 
the  detail  drawings,  and  that  is  the  machine  itself  in  operation. 


74  MACHINE  DESIGN 

Detail  drawings,  however,  are  infinitely  better  prosecutors  or 
defendants  than  a  multitude  of  wordy  counsel. 

Summary.  The  above  classification  of  machinery  might  be 
subdivided  and  extended  indefinitely,  and  on  the  broad  basis  on 
which  it  is  given  it  doubtless  does  not  cover  the  entire  field.  As 

o 

an  illustration,  however,  not  only  of  types  of  machinery,  but  of 
methods  of  design  and  study,  it  is  hoped  that  it  may  be  of  assist- 
ance in  giving  a  start  to  the  student  of  machine  design,  in  what- 
ever class  his  interests  may  happen  to  lie. 

It  is  the  general  principles  of  the  art  which  it  is  important  to 
master.  It  is  not  the  designing  of  a  locomotive,  or  a  stationary 
steam  engine,  or  a  crane,  or  an  engine  lathe,  or  a  rolling  mill, 
which  should  be  sought  to  be  learned,  but  the  designing  of  any- 
thing that  may  confront  us.  Specializing  is  sure  to  come  to  the 
designer  in  the  course  of  his  experience,  and  when  it  does  he  merely 
fits  to  the  particular  specialty  the  principles  he  knows  for  all,  and 
practically  develops  them  along  that  individual  line. 


. 


a 

•S    13 


t/1 

II 

o  o 
o 


MACHINE  DESIGN, 

PART  II. 


Introduction.  In  Part  I  is  illustrated  a  definite  and  systematic  method 
of  attacking  the  design  of  a  machine  as  a  whole.  In  Part  II  the  same  plan  is 
followed  with  regard  to  the  detail  of  its  component  parts,  the  machine  ele- 
ments which  are  chosen  as  illustrations  of  the  method,  being  the  simplest  and 
most  familiar  forms  in  common  use. 

As  before,  the  student  must  strive  to  grasp  and  absorb  the  method  of  design 
rather  than  any  specific  and  established  form  of  a  machine  part.  Part  II  is 
not  a  compendium  of  design,  does  not  attempt  to  be  complete  or  exhaustive  in 
any  of  its  chapters,  but  is  condensed  and  simplified  in  order  to  lead  the 
student  into  systematic  mechanical  thinking  and  logical  and  definite  action. 
Each  chapter  is  intended  to  stimulate  to  further  and  more  exhaustive  study 
along  lines  broader  than,  and  under  conditions  different  from  those  that  can 
be  specified  in  a  general  discussion.  But  no  matter  how  deeply  investigation 
may  be. carried,  or  how  specialized  the  study  may  become,  the  student  must 
realize  that  his  path  of  action  in  any  case  whatsoever  must  lie  along  the 
lines  of  Analysis,  Theory,  and  Practical  Modification  systematically  applied. 


BELTS. 

NOTATION — The  following  notation  is  used  throughout  the  chapter  on  Belts : 

A=  Sectional  area  of  belt  (square  inches)  R  =  Radius  of  pulley  (feet). 

=  6ft.  r  =  Radius  of  pulley  (inches). 

6= Width  of  belt  (inches).  T  =  Initial  tension  (Ibs.). 

F=Force  of  friction  at  pulley  rim  (Ibs.).  Tn=Total  tension  on  tight  side  (Ibs.).' 

h  =Thickness  of  belt  (inches)  T0=Total  tension  on  slack  side  (Ibs.). 

fJL= Coefficient  of  friction.  t   -  Working  tension  of  belt    (Ibs.  per 
N=Number  of  revolutions  of  pulley  per  square  inch). 

minute.  V  =  Velocity  of  belt  (feet  per  minute). 

n  =Fraction  of  circumference  of  pulley  w  = Weight  of  belt  per  cubic  inch  (Ibs.). 

embraced  by  belt.  2    =  Factor  due  to  centrifugal  force. 
P=Driving  force  at  pulley  rim  (lbs.)=F. 

ANALYS  IS.  When  a  belt  is  stretched  over  a  pair  of  pulleys, 
is  cut  off  at  the  proper  length,  and  is  laced  together  into  an  end- 
less band,  it  is  evident  that  as  long  as  the  belt  is  at  rest  there  is  a 
nearly  uniform  tension  in  it  throughout  its  length,  due  to  the  tight- 
ness with  which  the  lacing  is  drawn  up.  If  the  distance  between 
the  pulleys  is  considerable,  the  weight  of  the  belt  itself  as  it  hangs 
between  the  pulleys  will  produce  a  slightly  greater  tension  next  to 


76  MACHINE  DESIGN 

the  pulleys  than  exists  in  the  middle  of  the  span.  This  increase 
of  tension  due  to  the  weight  of  the  belt  would  make  but  little  dif- 
ference in  the  unit-stress  in  the  material  of  which  the  belt  is  made; 
hence  it  may  safely  be  assumed  that  the  tension  in  the  belt  when 
at  rest  is  uniform  throughout  its  entire  length. 

When  we  start  to  transmit  power  through  the  belt  by  turning 
one  of  the  pulleys,  thereby  driving  the  other  pulley  the  condition 
of  stress  in  the  belt  is  at  once  materially  changed.  As  the  belt  is 
a  flexible  member,  we  can  transmit  only  a  pull  to  the  other  pulley, 
thereby  turning  it  around,  the  push  which  is  at  the  same  time 
given  to  the  other  side  of  the  belt  merely  acting  to  make  the  belt 
sag  or  become  slack.  Hence  the  immediate  effect  of  starting  mo- 
tion in  a  belt  is  to  change  the  condition  of  equal  tension  through- 
out its  length,  to  that  of  unequal  tension  in  the  two  sides.  The 
driving  side  is  tight,  while  the  other  is  loose,  the  former  having 
gained  as  much  tension  as  the  latter  has  lost,  and  the  sum  of  the 
two  being  practically  equal  to  the  sum  of  the  tensions  in  the  two 
sides  of  the  belt  when  at  rest.  This  is  not  strictly  true,  as  will  be 
shown  later;  but  it  is  sufficiently  accurate  to  form  a  good  basis 
for  the  practical  design,  at  least  of  slow-speed  belts. 

.  This  condition  of  tight  and  slack  sides  is  made  possible  by 
the  fact  that  the  belt,  in  being  wrapped  around  the  pulleys  under 
tension,  has  friction  on  their  surfaces.  Thus,  we  can  pull  hard  on 
one  side  without  slipping  the  belt  around  the  pulleys,  but  could 
not  do  this  if  the  pulleys  were  perfectly  smooth  or  frictionless,  for 
in  that  case  the  elightest  pull  on  one  side  would  slip  the  belt 
around  the  pulleys.  In  fact,  it  W7ould  be  impossible  to  produce 
any  pull  by  means  of  the  driving  pulley,  for  the  pulley  would 
merely  slip  around  inside  the  belt. 

The  amount  of  pull  we  can  apply  to  the  belt  is  therefore  lim- 
ited by  the  tension  at  which  the  belt  slips  around  the  pulley. 
Moreover,  since  the  force  of  friction  between  the  belt  and  pulley 
is  dependent  upon  the  normal  force  with  which  the  belt  is  pressed 
against  the  pulley,  and  the  coefficient  of  friction  between  the  two, 
it  is  evident  that  the  tighter  the  belt  is  laced  up,  and  the  rougher 
the  surfaces  of  the  pulley  and  belt,  the  greater  is  the  force  that 
can  be  transmitted  through  the  belt.  This  leads  to  the  conclusion 
that  it  would  be  possible  to  transmit  any  amount  of  power  through 


MACHINE  DESIGN  77 


any  belt    however   small,    if    the   belt  were   only   laced    up  tight 
enough. 

This  conclusion  is  literally  true;  but  the  important  factTnow 
comes  in,  that  the  strength  of  the  material  of  which  the  belt  is 
made  is  limited,  and  while  theoretically  we  might  be  able  to  ac- 
complish the  above,  it  would  be  impossible  to  do  so  in  practice, 
for  at  a  certain  point  the  belt  would  break  under  the  strain.  Other 
practical  considerations  also  come  in,  which  fix  this  limit  of  power 
transmission  at  a  point  far  below  the  breaking  strength  of  thd  ma- 
terial. 

The  complete  analysis  is  not  quite  as  simple  as  the  above,  es- 
pecially for  high-speed  belts.  When  the  driving  side  of  the  belt 
becomes  tight,  it  stretches  and  grows  longer;  and  at  the  same 
time  the  other  side  of  the  belt  becomes  slack  and  grows  shorter. 
But  it  is  not  true  that  the  increase  in  the  one  side  is  the  same  as 
the  decrease  in  the  other,  and  this  fact  produces  the  condition  that 
the  sum  of  the  tensions  in  motion  is  not  quite  the  same  as  the  sum 
of  the  tensions  at  rest. 

Again,  when  the  belt,  as  it  passes  around  the  pulley,  changes 
its  straight-line  direction  to  circular  motion,  each  particle  of  the 
belt — like  a  body  whirling  at  the  end  of  a  cord  about  a  cehter  of 
rotation — tends  by  centrifugal  force  to  fly  away  from  the  surface 
of  the  pulley,  thereby  decreasing  the  normal  pressure,  and  hence 
the  friction.  This  centrifugal  force  also  changes  somewhat  the 
tensions  in  the  belt  between  the  pulleys.  As  the  centrifugal  force 
increases  in  proportion  to  the  square  of  the  linear  velocity,  it  is 
evident  that  the  effect  is  greater  at  high  speeds  than  at  moderate 
or  IOWT  speeds. 

A  further  circumstance  that  affects  the  driving  power  of  a 
belt  is  the  stiffness  of  the  leather  or  other  material  of  which  the 
belt  is  made.  As  it  passes  around  the  pulley,  the  belt  is  bent  to 
conform  to  the  circumference  of  the  pulley,  and  is  again  straight- 
ened out  as  it  leaves  the  pulley.  Hence  the  theoretically  perfect 
action  is  modified  somewhat  according  to  the  sharpness  of  the 
bending  and  the  thickness  or  flexibility  of  the  belt;  in  other  words,, 
a  small  pulley  carrying  a  thick  belt  would  be  the  worst  case  for 
successful  calculation  on  a  theoretical  basis. 

THEORY.     The  condition  of  the  tight  and   loose   sides  of  a 


78 


MACHINE  DESIGN 


belt  transmitting  power,  is  similar  to  that  of  the  weighted  strap 
and  fixed  pulley  shown  in  Fig.  17.  If  motion  is  desired  of  the 
strap  around  the  pulley,  it  is  necessary  to  make  the  weight  W2  of 
such  a  magnitude  that  it  will  overcome  not  only  the  weight  "W,, 
but  also  the  friction  between  the  strap  and  the  pulley.  The  strap 
tension  Tn  is,  of  course,  equal  to  W2,  and  T0  to  W,.  The  equation 
showing  the  balance  of  forces  for  the  condition  when  motion  is 
about. to  occur,  is: 

Tn  _  To  =  F  =  P  (.driving  force).  (5) 

If  the  pulley  be  free  to  turn  on  its  axis,  instead  of  being  fixed 

as  in  Fig.  17,  the  strap  by  its 
friction  on  the  pulley  will  turn 
the  pulley,  and  the  force  of 
friction  F  becomes  the  driving 
force  for  the  pulley  as  noted 
in  equation  5  above. 

In  Fig.  18,  let  us  sup- 
pose that  W  is  a  weight  repre- 
senting the  resistance  to  be 
overcome.  The  tensions  Tn 
and  T0,  equal  at  first  owing  to 
stretching  the  belt  tightly 
over  the  pulleys  at  rest,  change 
when  an  attempt  is  made  to 
raise  the  weight  by  turning 
the  larger  pulley;  and  just  as 
the  weight  leaves  the  floor,  the 
equality  of  moments  about 
the  axis  of  the  driven  pulley 
gives  the  following  equation: 

(Tu  -  T0)  r  =  F  X  r  =  P  X  r  ==  W  X  r, .          (6) 

This  equality  of  moments  remains  as  long  as  the  motion  of 
the  weight  is  uniform,  and  represents  closely  the  conditions  under 
which  belt  pulleys  work. 

Although  we  know  from  the  above  what  the  difference  of  the 
belt  tensions  is,  and  what  this  difference  will  do  when  applied  to 


Fig.  17. 


MACHINE  DESIGN 


79 


the  surface  of  a  given  pulley,  we  do  not  yet  know  what  either 
Tn  or  T0  actually  is;  and  until  we  do  know,  we  cannot  correctly 
proportion  the  belt.  Hence  we  must  find  another  relation  between 
Tn  and  T0  which  we  can  combine  with  equations  5  and  6.  This 
relation  is  deduced  by  a  process  of  higher  mathematics,  which  re- 
sults as  follows: 

'    T 

Common  logarithm    ~-=  2.729  p,  (1  -  z)n.       (7) 

-*-o 

Treating  equations  5  and  7  as  simultaneous,  values  of  both 
Tn  and  T0  can  be  found  by  the  regular  algebraic  solution.  As  Tn 
is  the  larger,  the  actual  area  of  belt  to  provide  the  necessary  strength 
must  be  made  to  depend  upon  it. 

The  factors  in  equation  7  depends  upon  the  centrifugal  force 


DRIVER. 


Fig.  18. 

developed  by  the  weight  of  the  belt  passing  around  the  pulley.     Its 
value,  found  from  mechanics,  is: 

w  X  V2 


z  = 


9,660  X^ 

Having  found  the  maximum  pull  on  the  belt,  it  now  remains 
to  write  the  equation: 


or, 


External  force  —  Internal  resistance; 
Tn  =  b  X  h  X  t. 


(8) 


Usually  the  most  convenient  way  to  handle  this  equation  is 
to  assume  h  and  t,  and  then  solve  for  b. 


80  MACHINE  DESIGN 

Summing  Tip  the  theoretical  treatment  of  belt  design,  we 
simply  combine  equations  5,  6,  7,  and  8,  and  solve  for  the  quantity 
desired.  Discussion  of  the  constants  involved  in  these  equations, 
and  of  the  practical  factors  controlling  them,  is  given  in  the  fol- 
lowing : 

PRACTICAL  MODIFICATION.  The  force  of  friction  F,  which 
is  the  same  as  driving  force  P,  depends  on: 

Coefficient  of  friction  (p.)  between  belt  and  pulley; 

Tightness  of  the  belt; 

Centrifugal  force  of  the  belt;    . 

Angle  of  contact  of  belt  with  pulley. 

The  coefficient  of  friction  (JLL),  according  to  experiments  and 
observed  operation  of  belts  transmitting  power,  varies  from  .15  to 
.56  for  leather  on  cast  iron.  An  average  value  consistent  with  a 
reasonable  amount  of  slip,  the  belt  being  in  good  running  order, 
is  .30.  If  the  belt  is  oily,  or  likely  to  become  so  in  use,  a  lower 
value  should  be  taken. 

The  tighter  the  belt  is  drawn  up,  the  greater  is  the  pressure 
against  the  pulley,  and  hence  the  greater  is  the  force  of  friction. 
But  if  we  pull  the  belt  up  too  tightly,  when  we  begin  to  drive, 
Tn  becomes  too  great,  and  the  belt  breaks  or  is  under  such  .stress 
that  it  wears  out  quickly.  Moreover,  the  great  side  pressure  on 
the  bearings  carrying  the  shaft  produces  excessive  friction,  and  the 
drive  is  inefficient.  This  is  why  a  narrow  belt  driven  at  high 
speed  is  more  efficient  than  a  wide  belt  at  slow  speed,  for  we  can- 
not pull  up  the  former  as  tightly  as  the  latter  without  overstraining 
it,  and  yet  it. is  possible  to  get  the  required  power  out  of  the  nar- 
row belt  by  running  it  at  high  speed. 

The  centrifugal  force  is  of  small  importance  for  low  speeds, 
say  of  3,000  feet  per  minute  and  less;  and  it  therefore  may  usu- 
ally be  neglected.  The  factor  s  then  becomes  zero  in  the  expres- 
sion 1  -  z  in  equation  7,  and  the  second  member  of  the  equation 
stands  simply  2.729  X  /x  X  n. 

The  angle  of  contact  of  belt  with  pulley  is  important,  as  a 
large  value  gives  a  great  difference  between  Tn  and  T0;  and  it  is 
desirable  to  make  this  difference  as  great  as  possible,  because  there- 
by the  driving  force  is  increased.  The  loose  side  of  a  horizontal 
belt  should  always  be  above,  as  then  the  natural  sag  of  the  loose 


MACHINE  DESIGN  81 


side  due  to  its  slackness  tends  to  increase  the  angle  of  contact  with 
the  pulley  y  while  the  tightening  up  of  the  lower  side  acts  against 
its  sag  to  make  the  loss  of  wrap  as  little  as  possible.  Vertical  belts 
which  have  the  driving  pulley  uppermost,  utilize  the  weight  of  the 
belt  to  increase  the  pressure  against  the  surface  of  the  pulley,  slightly 
increasing  its  capacity  for  driving.  The  angle  of  contact  may 
be  artificially  increased  by  a  tightening  pulley  which  presses  the 
belt  further  around  the  pulley  than  it  would  naturally  lie.  It 
adds  however,  the  friction  of  its  own  bearing,  and  impairs  the  effi- 
ciency of  the  drive.  For  ordinary  horizontal  belts,  the  angle  of 
contact  is  but  little  more  than  180°,  and  the  value  of  n  in  equation 
7  may  be  safely  assumed  at  |  unless  the  pulleys  are  of  relatively 
great  difference  of  diameter  and  very  close  together. 

Strength  of  Leather  Belting.  The  breaking  tensile  strength 
of  leather  belting  varies  from  3,000  to  5,000  pounds  per  square 
inch.  Joints  are  made  by  lacing,  by  metal  fasteners,  or  by  cement- 
ing. The  strength  of  a  laced  joint  may  be  about  T7T,  of  a  metal- 
fastened  joint,  about  |,  and  of  a  cemented  joint,  about  equal  to 
the  full  strength  of  the  belt  cross -sectional  area.  The  proper 
working  strength  of  belting  depends  on  the  use  to  which  the  belt 
is  put.  A  continuously  running  belt  should  have  a  low  tension 
in  order  to  have  long  life  and  a  minimum  loss  of  time  for  repairs. 
For  double  leather  belting  it  has  been  shown  that  a  working  ten- 
sion of  240  pounds  per  square  inch  of  sectional  area  gives  an  an- 
nual cost  —  for  repairs,  maintenance,  and  renewals  —  of  14  per 
cent  of  first  cost.  At  400  pounds  working  tension,  the  annual  ex- 
pense becomes  37  per  cent  of  first  cost.  These  results  apply  to 
belts  running  continuously;  larger  values  may  be  used  where  the 
full  load  comes  on  but  a  short  time,  as  in  the  case  of  dynamos. 

Good  average  values  for  working  tensions  of  leather  belts  are: 

Cemented  joints,  400  pounds  per  square  inch. 
Laced  joints,         300     "  "       " 

Metal  joints,          250     "  "       «         « 

Horse=Power  Transmitted  by  Belting.  If  P  is  the  driving 
force  in  pounds  at  the  rim  of  the  pulley,  and  V  is  the  velocity  of 
the  belt  in  feet  per  minute,  the  theoretical  horse-power  transmitted 
is  evidently  : 


82  MACHINE  DESIGN 


X  V 


--  ~3)00  ' 

It  is  evident  from  the  above  that  the  horse-power  of  a  belt  de- 
pends upon  two  things,  the  driving  force  P  and  the  velocity  Y.  If 
either  of  these  factors  is  increased,  the  horse-power  is  increased. 
Increasing  P  means  a  tight  belt.  Hence  a  tight  belt  and  high 
speed  together  give  maximum  horse-power.  But  a  tight  belt 
means  more  side  strain  on  shaft  and  journal.  Therefore,  from  the 
standpoint  of  efficiency,  use  a  narrow  belt  under  low  tension  at  as 
high  a  speed  as  possible. 

Empirical  rules  for  horse-power  of  belting,  if  used  with  judg- 
ment, give  safe  results  when  applied  to  very  general  cases.  A 
common  rule  used  by  American  engineers  is: 

;     fl-p-=W-  .       ('<»     ' 

For  a  double  belt,  assuming  double  strength,  this  becomes: 


With  large  pulleys  and  moderate  velocities,  this  may  hold 
good.  With  small  pulleys  and  high  velocities,  however,  the  un- 
certain stresses  induced  by  the  bending  of  the  fibers  of  the  belt 
around  the  pulley,  and  the  relatively  great  loss  due  to  centrifugal 
force,  modify  this  relation  •  and  a  safer  value  for  a  double  belt  of 
the  ordinary  kind  is: 


or,  still  safer,  H.  P.  =  (13) 


If  we  compare  the  theoretical  value  of  equation  9  with  the 
empirical  value  of  equation  10  by  putting  them  equal  to  each 
other,  thus: 

TT    p        PX  YSXV 

~3 

and  solve  for  P,  we  get  : 


MACHINE  DESIGN  83 

P  =  33J.  (i4) 

This  develops  the  fact  that  the  empirical  rule  of  equatrnn-10  as- 
sumes a  driving  force  of  33  pounds  per  inch  of  width  of  single 
belt. 

Another  way  of  expressing  equation  10  is:  A  single  belt 
will  transmit  one  horse-power  for  every  inch  of  width  at  a  belt 
speed  of  1,000  feet  per  minute. 

Speed  of  Belting.  The  most  economical  speed  is  somewhere 
between  4,000  and  5,000  feet  per  minute.  Above  these  values 
the  life  of  the  belt  is  shortened;  also  "flapping,"  "chasing,"  and 
centrifugal  force  cause  considerable  loss  of  power.  The  limit  of 
speed  with  cast-iron  pulleys  is  fixed  at  the  safe  limit  for  bursting 
of  the  rim,  which  may  be  taken  at  one  mile  per  minute. 

Material  of  Belting.  Oak- tanned  leather,  made  from  the 
part  of  the  hide  which  covers  the  back  of  the  ox,  gives  the  best  re- 
sults for  leather  belting.  The  thickness  of  the  leather  varies 
from  .18  to  .25  inch.  It  weighs  from  .03  to  .04  pound  per  cubic 
inch.  The  average  thickness  of  double  leather  belts  may  be  taken 
as  .33  inch,  although  a  variation  in  thickness  from  ^  inch  to  T7^ 
inch  is  not  uncommon.  Double  leather  belts  may  be  ordered 
light,  medium,  or  heavy. 

In  a  single- thickness  belt  the  grain  or  hair  side  should  be 
next  to  the  pulley,  for  the  flesh  side  is  the  stronger  and  is  there- 
fore better  able  to  resist  the  tensile  stress  due  to  bending  set  up 
where  the  belt  makes  and  leaves  contact  with  the  pulley  face. 
Double  leather  belts  are  made  by  cementing  the  flesh  sides  of 
two  thicknesses  of  belt  together,  leaving  the  grain  side  exposed 
to  surface  wear. 

Raw  hide  and  semi-raw  hide  belts  have  a  slightly  higher  co- 
efficient of  friction  than  ordinary  tanned  belts.  They  are  useful  in 
damp  places.  The  strength  of  these  belts  is  about  one  and  one- 
half  times  that  of  tanned  leather. 

Cotton,  cotton -leather,  rubber,  and  leather  link  belting  are 
some  of  the  forms  on  the  market,  each  of  which  is  especially 
adapted  to  certain  uses.  For  their  weights  and  their  tensile  and 
working  strengths  consult  the  manufacturers'  catalogues: 

A  prominent  manufacturer's  practice  in  regard  to  the  sizes  of 


84 


MACHINE  DESIGN 


leather  belting  will  be  found  useful  for  comparison,  and  is  indicated 
in  the  table  on  page  12. 

Initial  Tension  in  Belt.  On  the  assumption  that  the  sum  of 
the  tensions  is  unchanged,  whether  the  belt  be  at  rest  or  driving, 
we  should  have  the  following  relation  : 


whence, 


T    -I-  T  —  2T  • 

xn     i     -"-o  —  '°-L  5 

T  4-T 

m -^n   r  -ip 


(15) 


This  is  not  strictly  true,  however,  as  is  stated  in  the  "  Analysis '' 
of  "  Belts."  It  has  been  found  that  in  a  horizontal  belt  working  at 
about  400  Ibs.  tension  per  square  inch  on  the  tight  side,  and  hav- 
ing 2  per  cent  slip  on  cast-iron  pulleys  ( i.  e.,  the  surface  of  the 


Sizes  of  Leather  Belting. 


WIDTH. 

THICKNESS. 

Single. 

Double. 

1  inch. 

-jV  inch. 

T5u  inch. 

2     " 

A  " 

T5,      " 

3    " 

A  " 

f       " 

4    " 

A  " 

f       " 

5     " 

A  " 

1       " 

6    " 

A  " 

1       " 

10    " 

A  " 

t      ." 

12    " 

I       " 

14    " 

if    " 

20    " 

A    " 

driven  pulley  moving  2  per  cent  slower  than  that  of  the  driver), 
the  increase  of  the  sum  of  the  tensions  when  in  motion  over  the 
sum  of  the  tensions  at  rest,  may  be  taken  at  about  J  the  value  of 
the  tensions  at  rest.  Expressing  this  in  the  form  of  an  equation 


T  = 


(Tn+T0). 


(16) 


MACHINE  DESIGN  85 

The  value  of  T  thus  found  would  be  the  pounds  initial  tension  to 
which  the  belt  should  be  pulled  up  when  being  laced,  in  order  to 
produce  Tn  and  T0  when  driving. 

This  value  is  not  of  very  great  practical  importance,  as  the 
proper  tightness  of  belt  is  usually  secured  by  trial,  by  tightening 
pulleys,  by  pulley  adjustment  (as  in  motor  drives),  or  .by  shorten- 
ing the  belt  from  time  to  time  as  needed.  It  is  worth  noting, 
however,  that  for  the  most  economical  life  of  the  belt  it  would  be 
very  desirable  in  every  case  to  weigh  the  tension  by  a  spring  bal- 
ance when  giving;  the  belt  its  initial  tension.  This,  however,  is 

O  O  " 

not  always  easy  or  even  feasible;  hence  it  is  a  refinement  with 
which  good  practice  usually  dispenses,  except  in  the  case  of  large 
and  heavy  belts. 

PROBLEMS  ON   BELTS. 

1.  Determine  the  belt  tensions  in  a  laced  belt  transmitting  50 
horse-power  at  a  velocity  of  3,500  feet  per  minute.     Suppose  that 
the  arc  of  contact  is  180°;  weight  of  belt  —  .035  pound  per  cub. 
in.;  and  coefficient  of  friction  25  per  cent. 

2.  What  is  the  width  of  above  belt  if  it  is  T^  inch  in  thick- 
ness ? 

3.  What  initial  tension  must  be  placed  on  above  belt  ? 

4.  The  main  drive  pulley  of  a  120-horse-power  water  wheel 
is  6  feet  in  diameter.     A  cemented  leather  belt  is  to  connect  the 
main  pulley  to  a  3-foot  pulley  on  the  line  shafting  in  a  mill.    The 
horizontal  distance  between  centers  of  shafting  is  24  feet:  coeffi- 

O  ' 

cient  of  friction,  30  per  cent;  revolutions  per  minute  of  line  shaft- 
ing, 180.  Design  the  belt  for  this  drive. 

5.  An  8 -inch  double  belt  f  inch  thick  connects  2  pulleys  of 
30-inch  and  20-inch  diameter  respectively.     The  horizontal  dis- 
tance between  the  centers  is  12.5  feet.     The  coefficient  of  friction 
is  0.3,  and  the  weight  of  belt  per  cubic  inch  is  0.035  pound. 
Working  tension,  300  pounds  per  square  inch.     Speed  of  belt 
5,000  feet  per  minute.     Lower  face  of  30-inch  pulley  is  the  driv- 
ing face.     Required  the  H.  P.  which  may  be  transmitted  (theo- 
retically). 

6.  Compare  the  theoretical  horse-power  in  problem  5  with 
that  obtained  by  the  use  of  empirical  formula. 


86  MACHINE  DESIGN 

PULLEYS. 

NOTATION— The  following  notation  is  used  throughout  the  chapter  on  Pulleys: 

A  =Area  of  rim  (sq.  in.).  I    =  Length  of  hub  (inches). 

a=     "     "arm("      "  ).  N  =  Number  of  arms. 

b  =  Center  of  pulley  to  center  of  belt  n  =       "        "  rim  bolts,  each  side. 

(inches;  practically  equal  to  R).  P  =  Driving  force  of  belt  (Ibs.). 

GI  =  Total  centrifugal  force  of  rim  (Ibs.).  PI = Force   at   circumference   of     shaft 
c    =  Distance  from  neutral  axis  to  outer  (Ibs.). 

fiber  (inches).  Po=/Force  at  circumference  of  hub  (Ibs.). 

D  =  Diameter  of  pulley  (inches).  p  =  Stress  in  rim  due  to  centrifugal  force 
DI=        "           "hub       (     "     ).  (Ibs.  per  sq.  in.). 

d\  =       "  "  bolt  at  root  of  thread  K  =  Radius  of  pulley  (inches). 

(inches).  S   =  Fiber  stress  (Ibs.  per  sq.  in.). 

d  =  Diameter  of  bolt  holes  (inches).  s  = Fiber  stress  in  flange  (Ibs.  per  sq.  in.). 

g   =  Acceleration    due    to     gravity    (ft.  T  =  Thickness  of  web  (inches). 

per  sec.).  t    —  "  rim  (     "     ). 

h  =Widthof  arm  at  any  section  (inches).  ^2=  "  bolt  flange  (inches). 

I    =  Moment  of  inertia,  T  u=Tension  of  belt  on  tight  side  (Ibs.). 

L.  =  Length  of  arm,  center  of  belt  to  hub  T0=        "        "     "      "loose    "     (  "  ). 

(inches).  v   =  Velocity  of  rim  (ft.  per  sec.). 

Li=Length  of  rim  flange  of  split  pulley  w  =Weightof  material  (Ibs.  per  cub.  in.). 

(inches). 

ANALYSIS.  If  a  flexible  band  be  wrapped  completely  about 
a  pulley,  and  a  heavy  stress  be  put  upon  each  end  of  the  band,  the 
rim  of  the  pulley  will  tend  to  collapse  just  like  a  boiler  tube  with 
steam  pressure  on  the  outside  of  it.  A  compressive  stress  is  in- 
duced which  is  very  nearly  evenly  distributed  over  the  cross-sec- 
tion of  the  rim,  except  at  points  where  the  arms  are  connected 
thereto.  At  these  points  the  arms,  acting  like  rigid  posts,  take 
this  compressive  stress.  Now,  a  pulley  never  has  a  belt  wrapped 
completely  round  it,  the  fraction  of  the  circumference  embraced  by 
the  belt  being  usually  about  -|,  and  seldom,  even  with  a  tightener 
pulley,  reaching  |.  Assuming  the  wrap  to  be  \  the  circumference, 
and  that  all  the  side  pull  of  the  belt  comes  on  the  rim,  none  being 
transmitted  through  the  arms  to  the  hub,  we  then  have  one-half  of 
the  rim  pressed  hard  against  the  other  half  by  a  force  equal  to  the 
resultant  of  the  belt  tensions,  which,  in  this  case,  would  be  the 
sum  of  them.  Dividing  the  pulley  by  a  plane  through  its  center 
and  perpendicular  to  the  belt,  the  cross-section  of  the  rim  cut  by 
this  plane  has  to  take  this  compressive  stress- 

This  analysis  is  satisfactory  from  an  ideal  standpoint  only,  for 
the  intensity  of  stress  due  to  the  direct  pull  of  the  belt,  with  the 
usual  practical  proportions  of  rim,  would  be  very  small.  More- 
over, the  element  of  speed  has  not  been  considered. 

When  the  pulley  is  under  speed,  a  set  of  conditions  which 


MACHINE  DESIGN  87 

complicates  matters  is  introduced.  The  centrifugal  force  due  to 
the  weight  of  the  rim  and  arms  is  no  longer  negligible,  but  has 
an  important  influence  upon  the  design  and  material  ireeck  This 
centrifugal  force  acts  against  the  effect  of  the  belt  wrap,  tending 
to  reduce  the  compressive  stress,  or,  overcoming  the  latter  entirely, 
sets  up  a  tensional  stress  both  in  the  rim  and  in  the  arms.  It  also 
tends  to  distort  the  rim  from  a  true  circle  by  bowing  out  the  rim 
between  the  arms,  thus  producing  a  bending  moment  in  the  rim, 
maximum  at  the  points  where  the  rim  joins  each  arm. 

It  can  readily  be  imagined  that  the  analysis  in  detail  of  these 
various  stresses  in  the  rim  acting  in  conjunction  with  each  other 
is  quite  complicated  —  far  too  much  so  in  fact,  to  be  introduced 
here.  As  in  most  cases  of  such  design,  however,  one.  controll- 
ing influence  can  be  separated  out  from  the  others,  and  the  de- 
sign based  thereon  with  sufficient  margin  of  strength  to  satisfy 
the  more  obscure  conditions.  This  is  rational  treatment,  and  the 
"  theory  "  will  be  studied  accordingly. 

The  rim,  being  fastened  to  the  ends  of  the  arms,  tends,  when 
driving,  to  be  sheared  off,  the  resisting  area  being  the  areas  of  the 
cross-sections  of  the  arms  at  their  point  of  joining  the  rim.  The 
force  that  produces  this  shearing  tendency  is  the  driving  force  of 
the  belt,  or  the  difference  between  the  tensions  of  the  tight  and 
loose  sides. 

Again,  at  the  point  of  connection  of  the  arms  to  the  hub,  a 
shearing  action  takes  place,  so  that,  if  this  shearing  tendency  were 
carried  to  rupture,  the  hub  would  literally  be  torn  out  of  the  arms. 
Now,  viewing  the  arms  as  beams  loaded  at  the  end  with  the  driv- 
ing force  of  the  belt,  and  fixed  at  the  hub,  a  heavy  bending  stress 
ia  set  up,  which  is  maximum  at  the  point  of  connection  to  the 
hub.  If  the  rim  were  stiff  enough  to  distribute  this  driving  force 
equally  between  the  arms,  each  arm  would  take  its  proportional 
share  of  the  load.  The  rim,  however,  is  quite  thin  and  flexible; 
and  it  is  not  safe  to  assume  this  perfect  distribution.  It  is  usual 
to  consider  that  one-half  the  whole  number  of  arms  take  the  full 
driving  force. 

THEORY — Pulley  Rim.  Evidently  it  is  practically  impossible 
to  make  so  thin  a  rim  that  it  will  collapse  under  the  pull  of  a  belt. 
As  far  as  the  theory  of  the  rim  is  concerned,  its  proportion  prob- 


88 


MACHINE  DESIGN 


ably  depends  more  upon  the  calculation  for  centrifugal  force  than 
upon  anything  else. 

In  order  to  separate  this  action  from  that  of  any  other  forces, 
let  us  suppose  that  the  rim  is  entirely  free  from  the  arms  and  hub, 
and  is  rotating  about  its  center.  Every  particle,  by  centrifugal 
force,  tends  to  fly  radially  outward  from  the  center.  This  condi- 
tion is  represented  in  Fig.  19.  The  tendency  with  which  one-half 
of  the  rim  tends  to  fly  apart  from  the  other  is  indicated  by  the 
force  C,;  and  the  relation  between  C,  and  the  small  radial  force  c 
for  each  unit-length  of  rim  can  readily  be  found  from  the  prin- 
ciples of  mechanics.  The  case  is  exactly  like  that  of  a  boiler  or  a 
thin  pipe  subjected  to  uniform  internal  pressure,  which,  if  carried 
to  rupture,  would  split  the  rim  along  a  longitudinal  seam. 


Fig.  19.  Fig.  20. 

The  tensile  stress  thus  induced  per  square  inch  can  be  found 
by  simple  mechanics  to  be: 


;  (i7) 

y 
or,  since  w  =  0.26  pound,  and  g  =  32.2  feet  per  second, 

p  =  0.097  v2     (  say  -^ )  ;  (18) 

and,  if  p  be  taken  equal  to  1,000  pounds  per  square  inch,  which  is 
as  high  as  it  is  safe  to  work  cast  iron  in  this  place, 

v  ==  100  feet  per  second.  (l9) 

This  shows  the  curious  fact  that  the  intensity  of  stress  in  the  rim 


MACHINE  DESIGN 


89 


is  directly  proportional  to  the  square  of  the  linear  velocity,  and 
wholly  independent  of  the  area  of  cross-section.  It  is  also  to  be 
noted  that  100  feet  per  second  is  about  the  limit  of  speed  for  cast- 
iron  pulleys  to  be  safe  against  bursting. 

If  we  wish  to  consider  theoretically  the  rim  together  with  the 
arms  as  actually  connected  to  it,  we  get  a  much  more  complicated 
relation.  This  condition  is  shown  in  Fig.  20,  where  the  rim,  ex- 
panding more  than  the  arms,  bulges  out  between  them.  This 
makes  the  rim  act  something  like  a  continuous  beam  uniformly 
loaded;  but  even  then  the  resulting  stress  is  not  clearly  defined  on 
account  of  the  variable  stretch  in  the  arms.  Investigation  on  this 
basis  is  not  needed  further  than  to  note  that  it  is  theoretically 
better,  in  the  case  of  a  split  pulley,  to  make  the  joint  close  to  the 
arms,  rather  than  in  the  middle  of  a  span. 

Pulley  Arms.  The  centrifugal  force  developed  by  the  rim 
and  arms  tends  to  pull  the  arms  from  the  hub.  On  the  belt  side, 
this  is  balanced  to  some  extent  by  the  belt  wrap,  which  tends  to 
compress  the  arm  and  relieve  the  tension.  On  the  side  away 
from  the  belt,  the  centrifugal  action  has 
full  play,  but  the  arm  is  usually  of  such 
cross-section  that  the  intensity  of  this  stress 
is  very  low.  It  fnay  safely  be  neglected. 

The  rim  being  very  thin  in  most  cases, 
its  distributing  effect  cannot  be  depended 
on,  hence  the  driving  force  of  the  belt  may 
be  taken  entirely  by  the  arms  immediately 
under  the  portion  of  the  belt  in  contact  with 
the  pulley  face.  For  a  wrap  of  180°  this 


Fig.  21. 


means  that  only  one-half  of  the  pulley  arms  can  be  considered  as 
effective  in  transmitting  the  turning  effort  to  the  hub.  Each  of 
these  arms  is  a  lever  fixed  at  one  end  to  the  hub  and  loaded  at  the 
other.  A  lever  of  this  description  is  called  a  "  cantilever"  beam, 
its  maximum  moment  existing  at  its  fixed  end.  The  load  that  each 

p 
of  these  beams  may  be  subjected  to  is-^>   and  therefore  the  maxi- 


mum external  moment  at  the  hub  is  - 


2PL 


BT 


From   mechanics  we 


90  MACHINE  DESIGN 

know  that  the  internal  moment  of  resistance  of  any  beam  section 

SI 
is  —  ,  and   that  equilibrium  of   the  beam   can   be   satisfied  only 

when  the  external  moment  is  equal  to  the  internal  moment  of  re- 
si  stance  of  the  beam  section.  Equating  these  two,  we  have: 

2PL        SI 

IT     "-•  (20> 

The  arms  of  a  pulley  are  usually  of  the  elliptical  or  segmental 
cross-section,  and   may  be  of  the  proportions   shown   in  Fig.  21. 

For  either  of  these  sections  the  fraction  —-is  approximately  equal 

to  0.0393A3.  For  convenience  (the  error  caused  being  on  the  safe 
side),  L  may  be  taken  as  equal  to  the  full  radius  of  the  pulley  R, 
whence 


in  which  S  may  be  from  2,000  to  2,250  for  cast  iron 

Taking  moments  about  the  center  of  the  pulley,  and  solving 
for  P2,  the  force  acting  at  the  circumference  of  the  hub,  we  have  : 

2PR      PD 


4PR 

P= 


The  area  of  an  elliptical  section  is  TT  times  the  product  of  the 
half  axes.     With  the  proportions  of  Fig.  21,  this  becomes: 

vrfi2 

a  =  wX  0.2k  x  0.5A  =  -  ^  (23) 

Equating  the  external  force  to  the  internal  shearing  resistance,  we 
have  : 


4PR        ,r/,«S8 
r     "TO" 

40  PR 


ND, 


MACHINE  DESIGN 


91 


in  which  the  shearing  stress  Ss  may  run  from  1,500  to  1,800  for 
cast  iron. 

Although  both  bending  and  shearing  stresses  as~caiculated 
above  exist  at  the  base  of  the  arms,  the  bending  is,  in  practically 
every  case,  the  controlling  factor  in  the  design  of  the  arms.  An 
arm-section  large  enough  to  resist  bending  would  have  a  very  low 
intensity  of  shear. 

If  the  number  of  arms  be  increased  indefinitely,  we  come  to 
a  continuous  arm  or  web,  in  which  the  bending  action  is  elimi- 
nated. It  may  still  shear  off  at  the  hub,  where  the  area  of  metal 
is  the  least,  at  minimum  circumference.  In  this  case  the  area 
under  shearing  stress  is  TrDjT;  and  the  force  at  the  circumference 
of  the  hub,  is 

PR     2PR 


Equating  external  force  to  in- 
ternal shearing  resistance,  we 
have  : 


2  PR 


or,        .  = 


(25) 


Fig.  22. 


Pulley  Hub.  As  in  the 
case  of  the  arms,  centrifugal 
force  does  not  play  much  part  in  the  design  of  the  hub  of  a  pulley. 
The  hub  is  designed  principally  to  carry  the  key,  and  through  it 
transmit  the  turning  moment  to  the  shaft.  Considered  thus,  the 
hub  may  tear  along  the  line  of  the  key  or  crush  in  front  of  the  key. 

For  example,  in  Fig.  22,  if  the  connection  with  the  lower 
arms  be  neglected,  and  the  upper  arms  be  held  fast  while  a  turning 
force  Pj,  at  the  surface  of  the  shaft,  is  transmitted  to  the  hub. 
through  the  key,  then  the  metal  of  the  hub  directly  in  front  of  the 
key  is  under  crushing  stress ;  and  the  metal  along  the  line  eb,  from 
the  corner  to  the  outside,  is  under  tensile  stress.  This  condition  is 
the  worst  that  could  possibly  happen,  because  the  bracing  effect  of 
the  lower  arms  has  been  neglected,  and  the  key  is  located  between 
the  arms. 


92  MACHINE  DESIGN 

Taking  moments  about  the  center  of  the  shaft,  the  value  of  the 
force  at  the  shaft  circumference,  or  the  "key  pull,"  is: 


P          & 

TJ^  =  —  ,  k  being  the  distance  from   the   center   of  shaft    to 

center  of  eb,  and  the  area  of  metal  which  is  subjected  to  the  tearing 
action  P3  is  I  X  eb.  Equating  the  external  force  to  the  internal 
resistance,  and  assuming  that  the  stress  is  equally  distributed  over 
the  area  I  X  eb,  we  have  : 


T>T> 

(27) 


•  k  X  I  X  eb 

The  intensity  of  crushing  on  the  metal  in  front  of  the  key,  due 
to  force  Pn  depends  upon  the  thickness  of  the  key,  and  is  properly 
discussed  later  under  "Keys." 

PRACTICAL  MODIFICATION— Pulley  Rim.  The  theoretical 
calculation  for  the  thickness  of  the  rirn  may  give  a  thickness  that 
could  not  be  cast  in  the  foundry,  and  the  section  in  that  case  will 
have  to  be  increased.  As  light  a  section  as  can  be  readily  cast  will 
usually  be  found  abundantly  strong  for  the  forces  it  has  to  resist. 
A  minimum  thickness  at  the  edge  of  the  rim  is  about  T36  inch; 
and  as  the  pulleys  increase  in  size,  the  rim  also  must  be  made 
thicker;  otherwise  the  rim  will  cool  so  much  more  quickly  than 
the  arms,  that  the  latter,  on  cooling,  will  develop  shrinkage  cracks 
at  the  point  of  junction. 

For  a  velocity  of  6,000  feet  per  minute,  we  find  from  equation 
18  that  the  tension  in  pounds  per  square  inch,  in  the  rim,  due  to 
centrifugal  force,  is  970.  Though  this  in  itself  is  a  low  value,  yet 
the  uncertain  nature  of  cast  iron,  its  condition  of  internal  stress, 
due  to  casting,  and  the  likely  existence  of  hidden  Haws  and  pockets, 
have  established  the  usage  of  this  figure  as  the  highest  safe  limit 
for  the  peripheral  speed  of  cast-iron  pulleys.  It  is  easily  remem- 
bered that  cast-iron  pulleys  are  safe  for  a  linear  velocity  of  about 
one  'mile per  minute. 


MACHINE  DESIGN 


To  prevent  the  belt  from  running  off  the  pulley,  a  "crown" 
or  rounding  surface  is  given  the  rim.  A  tapered  face,  which  is 
more  easily  produced  in  the  ordinary  shop,  may  be  nse4- -instead. 
This  taper  should  be  as  little  as  possible,  consistent  with  the  belt 
staying  on  the  pulley;  J  inch  per  foot  each  way  from  the  center 
is  not  too  much  for  faces  4  inches  wide  and  less;  while  above  this 
width  J  inch  per  foot  is  enough.  As  little  as  J-  inch  total  crown 
has  been  found  to  be  sufficient  on  a  24-inch  face,  but  this  is 
probably  too  little  for  general  service. 

Instead  of  being  "crowned,"  the  pulley  may  be  flanged  at  the 
edges;  but  flanged  pulley  rims  chafe  and  wear  the  edge  of  the  belt. 

The  inside  of  the  rim  of  a  cast-iron  pulley  should  have  a  taper 
of  -J  inch  per  foot  to  permit  easy  withdrawal  from  the  foundry 


Fig.  23. 

mould.  This  is  known  as  "draft."  If  the  pattern  be  of  metal,  or 
if  the  pulley  be  machine-moulded,  the  greater  truth  of  the  casting 
does  not  require  that  the  inside  of  the  rim  be  turned,  as  the  pulley, 
at  low  speeds,  will  be  in  sufficiently  good  balance  to  run  smoothly. 
For  roughly  moulded  pulleys,  and  for  use  at  high  speeds,  however, 
it  is  necessary  that  the  rim  be  turned  on  the  inside  to  give  the 
pulley  a  running  balance. 

Fig.  23  shows  a  plain  rim  a  also  one  stiffened  by  a  rib  b. 
Where  heavy  arms  are  used  this  rib  is  essential  so  that  there  will 
not  be  too  sudden  change  of  section  at  the  junction  of  rim  and  arm. 
and  consequent  cracks  or  spongy  metal. 

Pulley  Arms.  The  arms  should  be  well  filletted  at  both  rim 
and  hub,  to  render  the  flow  of  metal  free  and  uniform  in  the  mould. 
The  general  proportions  of  arms  and  connections  to  both  hub  and 
rim  may  perhaps  be  best  developed  by  trial  to  scale  on  the  draw- 
ing board.  The  base  of  the  arm  being  determined,  it  may  gradu- 


94  MACHINE  DESIGN 

ally  taper  to  the  rim,  where  it  takes  about  the  relation  of  §  to  | 
the  dimensions  chosen  at  the  hub.  The  taper  may  be  modified 
until  it  looks  right,  and  then  the  sizes  checked  for  strength. 

Six  arms  are  used  in  the  great  majority  of  pulleys.  This 
number  not  only  looks  well,  but  is  adapted  to  the  standard  three- 
jawed  chucks  and  common  clamping  devices  found  in  most  shops. 
Elliptical  arms  look  better  than  the  segmental  style.  The  flat, 
rectangular  arm  gives  a  very  clumsy  and  heavy  appearance,  and  is 
seldom  found  except  on  the  very  cheapest  work. 

A  double  set  of  arms  may  be  used  on  an  excessively  wide 
face,  but  it  complicates  the  casting  to  some  extent. 

Although  a  web  pulley  may  be  calculated  for  shear  at  the 
hub,  yet  it  will  usually  be  found  that  with  a  thickness  of  web  in- 
termediate between  the  thickness  of  the  rim  and  that  of  the  hub, 
which  will  satisfy  the  casting  requirements,  the  requirements  as  to 
strength  will  be  fully  met. 

Pulley  Hub.  The  hub  should  have  a  taper  of  ^  inch  per  foot 
draft,  similar  to  that  of  the  inside  of  the  rim.  The  length  of  the 
hub  is  arbitrary,  but  should  be  ample  to  prevent  rocking  on  the 
shaft.  A  common  rule  is  to  make  it  about  |  the  face  width  of 
the  pulley. 

The  diameter  of  the  hub,  aside  from  the  theoretical  consider- 
ation given  above,  must  be  sufficient  to  take  the  wedging  action  of  a 
taper  key  without  splitting.  This  relation  cannot  well  be  calcu- 
lated. Probably  the  best  rule  that  exists  is  the  familiar  one  that 
the  hub  should  be  twice  the  diameter  of  the  shaft.  This  rule, 
however,  cannot  be  literally  adhered  to,  as  it  gives  too  small  hubs 
for  small  shafts  and  too  large  ones  for  large  shafts.  It  is  always 
well  to  locate  the  key,  if  possible,  underneath  an  arm  instead  of 
between  the  arms,  thus  gaining  the  additional  strength  due  to  the 
backing  of  the  arm. 

SPLIT  PULLEYS. 

ANALYSIS  and  THEORY.  The  split  pulley  is  made  in 
halves  and  provided  with  bolts  through  flanges  and  bosses  on  the 
hub  for  holding  the  two  halves  together.  When  the  pulley  is  in 
place  on  the  shaft,  bolted  up  as  one  piece,  it  is  subjected  to  the 
same  forces  as  the  simple  pulley.  Hence  its  general  design  fol- 


BULLOCK  ELECTRIC  MANUFACTURING  Co. 

0)    <^r>  CINCINNATI, O.  U.  S.  A. 

§   <5jor*> 


P/n  on  this  fast 


So  ate;  j  Size 


9983-1 
J-3-S902 


from  l    ; 
length  over 
off,  of  GO 
cfiartyeof 
to  Sts't       ' 
VQ83 


99S3-2 


to 

ta/oer  pin 
A.B.W. 


$983-3 
4-/-/902 


Material. 
Tooi  No* 
Pat.  No 


SHAFT  AND  Ass  EMBLY    

OF    DRUMS  £st.  Weight 

7  CONTROLLER ...^ 

Draftsman.  ^A.B  JAi.^ 


EXAMPLE  OF  SHOP  DRAWING  REPRODUCED  FROM  A  BLUE  PRINT  BY  THE  COURTES\ 
OF  THE  BULLOCK  ELECTRIC  MFG.  CO. 


MACHINE  DESIGN 


95 


lows  the  same  principles,  and  we  need  only  study  the  fastening  of 
the  two  halves,  and  the  effect  of  this  fastening  on  the  detail  of  rim 
and  hub. 

The  simplest  stress  we  have  to  consider  on  the  rim  bolts  is 
one  of  pure  tension,  due  to  the  centrifugal  force  of  the  halves 
of  the  pulley,  A  safe  assumption  to  make  is  that  the  rim  is  free 


Fig.  24. 

from  the  arms  and  hub,  as  in  the  simple  pulley,  and  that  the  cen- 
trifugal force  developed  by  it  has  to  be  taken  by  the  rim  bolts 
alone.  In  other  words,  consider  the  rim  bolts  as  belonging  en- 
tirely to  the  rim,  and  make  them  as  strong  as  the  rim,  leaving  the 
hub  bolts  to  take  the  centrifugal  force  of  the  arms  and  hub,  and 
the  spreading  tendency  due  to  the  key. 

Another  tensile  stress  is  induced  in  the  rim  bolts  by  the  fact, 
that,  having  made  an  open  joint  in  the  rim,  and  in  addition  placed 
the  extra  weight  of  lugs  there,  the  centrifugal  action  at  this  point 
is  increased,  and  at  the  same  time  a  point  of  weakness  in  the  rim 


96 


MACHINE  DESIGN 


introduced.  Referring  to  Fig.  24,  the  rim  flanges  EJ  tend  to  fly 
out  due  to  the  centrifugal  force  CF.  This  tends  to  open  the  joint 
J  at  the  outside  of  the  rim  ;  to  throw  a  bending  stress  on  the  rim, 
maximum  at  the  point  E  ;  and  to  "heel"  the  rim  flanges  about 
the  point  E.  The  rim  bolts  acting  on  the  leverage  e  about  the 
point  E  must  resist  these  tendencies,  and  are  thereby  put  in 
tension. 

Referring  to  equation  18,  we  find  the  intensity  of  stress  due  to 
the  centrifugal  force  of  the  rim  in  Ibs.  per  square  inch  to  be  : 


V- "  TO" 

If  A  is  the  sectional  area  of  the  rim  in  square  inches,  this  means 
that  the  total  strength  of 
the  rim  is  represented    by 

A  rt«2 

— -.  The  strength  of  a 
bolt  is  represented  by  the 

bTT^j2. 

expression  — |—       if >  now> 

there  are  n  bolts  in  the 
flange,  the  total  resisting 

n$7Td* 
force  of  the  bolts  is  — ^— ; 

and  the  equation  represent- 
ing equality  of  strength  be- 
tween rim  and  bolts  is  : 


A?;2 


(28) 


from  which,  by  a  proper 
assumption  of  the  fiber 
stress  S,  which  should  be 

low,  the  opening-up  tendency  of  the  joint  being  neglected,  the  diam- 
eter at  the  root  of  the  thread  dl  may  be  calculated,  and  the  nom- 
inal bolt  diameter  chosen.  Reference  to  the  table  for  strength 
of  bolts,  given  in  the  chapter  on  Bolts,  Studs,  etc.,  will  be  found 
convenient. 


MACHINE  DESIGN  97 

It  is  very  doubtful  if  the  tension  on  the  flange  bolts,  due  to 
the  "  heeling  "  about  E  can  be  calculated  with  sufficient  accuracy  to 
be  of  much  value.  It  is  probably  better  to  assume  S  at  a~low  value, 
say  4,000,  and,  in  addition,  for  large  and  high-speed  pulleys,  stif- 
fen the  rim  by  running  a  rib  between  the  flange  and  the  adjacent 
arm.  It  is  evident  that  if  we  make  the  rim  so  stiff  that  it  cannot 
deflect,  there  will  be  no  u  heeling "  about  E  ;  and  the  bolts  will 
be  well  proportioned  by  the  preceding  calculation,  giving  them 
equal  strength  to  that  of  the  rim  section. 

For  the  bolt  flange  itself,  any  tendency  to  open  at  the  joint  J 
would  cause  it  to  act  like  a  beam  loaded  at  some  point  near  its 
middle  with  the  bolt  load,  and  supported  at  J  and  E.  This 
condition  is  shown  in  Fig.  25.  Probably  the  weakest  section 
would  be  along  the  line  of  the  bolt  centers.  We  have  just  noted 

that  the  carrying  capacity  of  the  bolts  is  -  -  .  '  .  Hence,  assum- 
ing that  e  =  \f,  which  is  about  the  worst  case  which  could  hap- 
pen, we  have  a  beam  of  length/' loaded  at  the  middle  with  - 

and  supported  at  the  ends.  Equating  the  external  moment  to 
the  internal  moment,  we  have  : 

n$7rd*       /       *(L,  -  nd}t?  ,   m 

~T~        T~:  ~6~ 

from  which  the  fiber  stress  6-  in  the  flange  may  be  calculated  and 
judged  for  its  allowable  value. 

L!  maybe  assumed  a  little  narrower  than  the  pulley  face;  and 
t2  from  1  inch  to  2  inches  or  more,  depending  on  the  thickness  of 
the  rim. 

The  hub  bolts  doubtless  assist  the  rim  bolts  in  preventing 
the  halves  of  the  pulley  from  flying  apart.  They  also  clamp  the 
hub  tightly  to  the  shaft,  preventing  any  looseness  on  the  key. 
Their  function  is  a  rather  general  one;  and  the  specific  stress 
which  they  receive  is  practically  impossible  to  calculate.  As  a 
matter  of  fact,  if  the  hub  bolts  were  left  out  entirely,  the  pulley 
would  still  drive  fairly  well,  but  general  rigidity  and  steadiness 
would  be  impaired.  Hence  the  size  of  the  hub  bolts  is  more  a 
practical  question  than  one  involving  calculation.  The  rim  bolts 


98  MACHINE  DESIGN 

should  be  figured  first,  and  their  size  determined  on ;  then  the  hub 
bolts  can  be  judged  in  proportion  to  the  rim  bolts,  the  diameter  of 
shaft,  the  thickness  and  length  of  the  hub,  and  the  general  form 
of  the  pulley.  Often  appearance  is  the  deciding  factor,  it  being 
manifestly  inconsistent  to  associate  small  fastenings  with  large 
shafts  or  hubs,  even  though  the  load  be  actually  small. 

PRACTICAL  MODIFICATION.  Practical  considerations  are 
chiefly  responsible  for  the  location  of  the  joint  in  a  split  pulley 
between  the  arms  instead  of  directly  at  the  end  of  an  arm,  where 
theoretically  it  would  seem  to  be  required.  It  is  usually  more 
convenient  in  the  foundry  and  machine  shop  to  have  the  joint  be- 
tween the  arms;  so  we  generally  find  it  placed  there,  and  strength 
"provided  to  permit  this.  It  is  possible,  however,  to  provide  a 
double  arm,  or  a  single  split  arm,  in  which  case  the  joint  of  the 
pulley  comes  at  the  arm,  and  the  "  heeling "  action  'of  the  rim 
flanges  is  prevented. 

The  rim  bolts  should  be  crowded  as  close  as  possible  to  the 
rim  in  order  to  reduce  the  stress  on  them,  and  also  the  stress  in 
the  flange  itself.  The  practical  point  must  not  be  forgotten,  how- 
ever that  the  bolts  must  have  sufficient  clearance  to  be  put  into 
place  beneath  the  rim. 

While  it  is  evident  that  the  rim  bolts  are  most  effective  in 
taking  care  of  the  centrifugal  action  of  the  halves,  yet  in  small 
split  pulleys  it  is  quite  common  to  omit  the  rim  bolts  and  to 
use  the  i:ub  bolts  for  the  double  purpose  of  clamping  the  shaft 
and  holding  the  two  halves  together.  The  pulley  is  cast  with  its 
rim  continuous  throughout  the  full  circle,  and  it  is  machined  in 
this  form.  It  is  then  cracked  in  two  by  a  "well -directed  blow  of  a 
cold  chisel,  the  casting  being  especially  arranged  for  this  along  the 
division  line  by  cores  so  set  that  but  a  narrow  fin  of  metal  holds 
the  two  parts  together.  This  provides  sufficient  strength  for  cast- 
ing and  turning,  but  permits  the  cold  chisel  to  break  the  connec- 
tion easily. 

SPECIAL  FORflS  OF  PULLEYS. 

The  plain  cast-iron  pulley  has  been  used  in  the  foregoing 
discussion  as  a  basis  of  design.  A  pulley  is,  however,  such  a 
common  commercial  article,  and  finds  such  universal  use,  that 


MACHINE  DESIGN  99 

special  forms,  which  can  be  bought  in  the  open  market,  are  not 
only  cheaper  but  better  than  the  plain  cast-iron  pulley,  at  least  for 
regular  line-shaft  work. 

Cast  iron  is  a  treacherous  and  uncertain  material  for  rims  of 
pulleys.  It  is  not  well  suited  to  high  fiber  stresses;  hence  the  range 
of  speed  permissible  for  pulley  rims  of  cast  iron  is  limited.  Steel 
and  wrought  iron,  having  several  times  the  tensional  strength  of 
cast  iron,  and  being,  moreover,  much  more  nearly  homogeneous 
in  texture,  are  well  suited  for  this  work;  one  of  the  best  pulleys  on 
the  market  consists  of  a  steel  rim  riveted  to  a  cast-iron  spider. 
Such  an  arrangement  combines  strength  and  lightness,  without 
increasing  complication  or  expense. 

The  all-steel  pulley  is  a  step  further  in  this  direction.  Here 
the  rim,  arms,  and  hub  are  each  pressed  into  shape  by  specially 
devised  machinery,  then  riveted  and  bolted  together.  This  pulley 
is  strictly  a  manufactured  article,  which  could  not  compete  with  the 
simpler  forms  unless  built  in  large  quantities,  enabling  automatic 
machinery  to  be  used.  Large  numbers  of  pulleys  are  built  in  this 
way,  and  are  put  on  the  market  at  reasonable  prices. 

Wood-rim  pulleys  have  been  made  for  many  years,  and, 
except  for  their  clumsy  appearance,  are  excellent  in  many  respects. 
The  rim  is  built  up  of  segments  in  much  the  same  way  as  an  ordi- 
nary pattern  is  made,  the  segments  being  so  arranged  that  they 
will  not  shrink  or  twist  out  of  shape  from  moisture.  The  hubs 
may  be  of  cast  iron,  bolted  to  wooden  webs,  and  carrying  hard- 
wood split  bushings,  which  may  be  varied  in  bore  within  certain 
limits  so  as  to  fit  different  sizes  of  shafting.  The  wooden  pulley 
is  readily  and  most  often  used  in  the  split  form,  thus  enabling  it 
to  be  put  in  position  easily  at  any  point  of  a  crowded  shaft.  It  is 
often  merely  clamped  in  place,  thus  avoiding  the  use  of  keys  or 
set  screws,  and  not  burring  or  roughening  the  shaft  in  any  way. 

PROBLEMS  ON  PULLEYS. 

1.  Calculate  the  tensile  stress  due  to  centrifugal    force  in 
the  rim  of  a  cast-iron  pulley  30  inches  in  diameter,  at  500  revolu- 
tions per  minute. 

2.  The  driving  force  of  a  belt  on  a  36-inch  pulley  is  800 
Ibs.,  and  the  belt  wrap  about  180°.     Calculate  proportions  of  el- 


100  MACHINE  DESIGN 


liptical  arms  to  resist  bending,  the  allowable  fiber   stress  being 
2,000. 

3.  A  pulley  12  inches  in  diameter,  |-inch  web,  4- inch  diam- 
eter hub,  transmits  25  horse-power  at  a  belt  speed  of  3,000  ft. 
per  minute.     Calculate  the  maximum  shearing  stress  in  the  web. 

4.  In  Fig.  24   assume  the  following  data:    L,  =  7  inches; 
^2=1  inch;    <?  — 1|-  inches \f=  3  inches;  area  of  rim  =  3  sq. 
in.;  allowable  tensile  stress  in  rim  1,000  Ibs.  per  sq.  in.    Calculate 
the  diameter  of  the  rim  bolts. 

5.  Calculate  the  fiber  stress  in  the  rim  bolt  flange  along  the 
line  of  the  bolts. 

SHAFTS. 

NOTATION— The  following  notation  is  used  throughout  the  chapter  on  Shafts  : 

Ao= Angular  deflection  (degrees).  L  =  Length  along  shaft  (inches). 

B  =Simple  bending  moment  (inch-lbs.).        LI,  L_>  =  Length  of  bearings  (inches.) 

Be=Equivalent  bending  moment  (inch-        M  ^Distance  between  bearings  (feet.) 

s'''  N  =Number  of  revolutions  per  minute. 

c    =  Distance  from  neutral  axis  to  outer 

fiber  finches)  P  =DrivinS  force  of  belt  <lbs->- 

iiucr  iinciics^  •  ..-*-.-        •*          •»-•«  ,     •»  /n     \ 

d,  d0,  d2,   d3,    d4=Diameters    of    shaft  Pi  =  Load  applied  as  stated  (Ibs.). 

(inches).  ^  =  Radius  at  which  load  as  stated  acts 

c?i=Internal  diameter  of  shaft  (inches).  (inches). 

E=  Direct     modulus    of     elasticity     (a  S  =  Fiber    stress,   tension,  compression, 

ratio),  or  shearing  (Ibs.  per  sq.  in.). 

e  =  Transverse  deflection  (inches).  T  =  Simple  twisting  moment  (inch-lbs.). 

G=Transverse  modulus  of  elasticity  (a  Te= Equivalent  twisting  moment  (inch- 
ratio).  lbs>)> 

H=Horse-power  (33,000  ft.-lbs.  per  min-  Tn=Tension  in  tight  side  of  belt  (Ibs.). 

ute).  T0=Tension  in  loose  side  of  belt  (Ibs.). 

I  =  Moment  of  inertia.  W=Load  applied  as  stated  (Ibs.). 

K=Distance  between  bearings  (inches). 

ANALYSIS.  The  simplest  case  of  shaft  loading  is  shown  in 
Fig.  26.  The  equal  forces  W,  similarly  applied  to  the  disc  at  the 
distance  E  from  its  center,  tend  to  twist  the  shaft  off,  the  tendency 
being  equal  at  all  points  of  the  length  L  between  the  disc  and  the 
post,  to  which  the  shaft  is  rigidly  fastened.  The  fastening  to  the 
post,  of  course,  in  this  ideal  case,  takes  the  place  of  a  resisting 
member  of  a  machine.  A  state  of  pure  torsion  is  induced  in  the 
shaft;  and  any  element,  such  as  ca,  is  distorted  to  the  position  c5, 
aob  being  the  angular  deflection  for  the  distance  L.  * 

The  case  of  Fig.  27  is  illustrative  of  what  occurs  when  a  belt 

o 

pulley  is  substituted  for  the  simple  disc.    Here  the  twisting  action 
is  caused  by  the  driving  force  of  the  belt,  which  is  Tn  -  T0  =  =  P, 


MACHINE  DESIGN 


101 


acting  at  the  radius  R.  Torsion  and  angular  deflection  exist  in 
the  shaft,  as  in  Fig.  26.  In  addition,  however,  another  stress  of 
a  different  kind  has  been  introduced;  for  not  only  does^the  shaft 
tend  to  be  twisted  off,  but  the  forces  Tn  and  TD ,  acting  together, 
tend  to  bend  the  shaft,  the  bending  moment  varying  with  every 
section  of  the  shaft,  being  nothing  at  the  point  0,  and  maximum 
at  the  point  c.  This  combined  action  is  the  most  common  of  any 
that  we  find  in  ordinary  machinery,  occurring  in  nearly  every  case 
with  which  we  have  to  deal. 

In  Fig.  27,  if  the  forces  Tn  and  T0  be  made  equal,  there  will 
be  no  tendency  at  all  to  twist  off  the  shaft,  but  the  bending  will 
remain,  being  maximum  at  the  point  c.  This  condition  is  illustra- 
tive of  the  case  of  all  ordinary  pins  and  studs  in  machines.  In 

this  sense,  a  pin  or  a  stud  is  sim- 
ply a  shaft  which  is  fixed  to  the 
frame  of  the  machine,  there  be- 
ing no  tendency  to  turning  of  the 
pin  or  stud  itself.  The  same 
condition  would  be  realized  if 
the  disc  in  Fig.  27  were  loose 
upon  the  shaft.  In  that  case, 
the  bending  moment  would  be 
caused  by  Tn  -f-  T0  acting  with 
the  leverage  L.  Of  course  there 
would  have  to  be  some  resistance 
for  Tn-T0  to  work  against,  in 
order  that  torsion  should  not  be 
transmitted  through  the  shaft. 
This  condition  might  be  intro- 
Fig.  26.  duced  by  having  a  similar  disc 

lock  with  the  first  one  by  means 
of  lugs  on  its  face,  thus  receiving  and  transmitting  the  torsion. 

If  the  distance  L  becomes  very  great,  both  the  angular  deflec- 
tion due  to  twisting,  and  the  sidewise  deflection  due  to  bending, 
become  excessive,  and  not  permissible  in  good  design.  This 
trouble  is  remedied  by  placing  a  bearing  at  some  point  closer  to 
the  disc,  which,  as  it  decreases  L,  of  course,  decreases  the  bending 
moment  and  therefore  the  transverse  deflection.  The  angular  de- 


102 


MACHINE  DESIGN 


flection  can  be  decreased  only  by  brinoincr  the  resistance   and  load 

J         J  O        O 

nearer  together. 

The  above  implies,  of  course,  that  the  diameter  of  the  shaft  is  not 
changed,  it  being  obvious  that  increase  of  diameter  means  increase  of  strength 
and  corresponding  decrease  of  both  angular  and  transverse  deflection. 

If  the  speed  of  the  shaft  be  very  high,  and  the  distance  be- 
tween bearings,  represented  by  L,  be  very  great,  the  shaft  will  take 
a  shape  like  a  bow  string  when  it  is  vibrated.,  and  smooth  action 
cannot  ba  maintained. 

It  is  necessary  to   carry  the  cases   of  Figs.   26    and  27  but  a 


Fig.  27. 

single  step  farther  to  illustrate  the  actual  working  conditions  of 
shafting  in  machines.  Suppose  the  rigid  post  to  have  the  shaft 
passing  clear  through  it,  and  to  act  as  a  bearing,  so  that  the  shaft 
can  freely  rotate  in  it,  the  resistance  being  exerted  somewhere  be- 
yond. The  twisting  moment  will  be  unchanged,  also  the  bending 
moment:  but  the  effect  of  the  bending  moment  will  be  on  each 

o 

particle  of  the  shaft  in  succession,  now  putting  compression  on  a 
given  particle,  and  then  tension,  then  compression  again,  and  so 
on,  a  complete  cycle  being  performed  for  each  revolution.  This 


MACHINE  DESIGN  %  103 

brings  out  a  very  important  difference  between  the  bending  stress 
in  pins  and  the  bending  stress  in  rotating  shafts.  In  the  one  case 
the  bending  stress  is  non -reversing;  in  the  other,  reversing;  and 
a  much  higher  fiber  stress  is  permissible  in  the  former  than  in  the 
latter. 

THEORY — Simple  Torsion.  In  the  case  of  simple  torsion 
the  stress  induced  in  the  shaft  is  a  shearing  one.  The  external 
moment  acU  about  the  axis  of  the  shaft,  or  is  a  polar  moment; 
hence  in  the  expression  for  the  moment  of  the  internal  forces,  the 
polar  moment  of  inertia  must  be  used.  Now,  from  mechanics  we 
have: 

I          d3 
and  -  =  -^-y-    (for  circular  section  of  diameter  d) ; 

therefore,  T  —  ~R~T~'  (3°) 

from  which  the  diameter  for  any  given  twisting  moment  and  fiber 
stress  can  readily  be  found. 

For  a  hollow  shaft  this  expression  becomes; 

T  = 


5.1r/0 

Simple  Bending.  The  stresses  induced  in  a  pin  or  shaft  under 
simple  bending  are  compression  and  tension.  The  external  moment 
in  this  case  is  transverse,  or  about  an  axis  across  the  shaft;  hence 
the  direct  moment  of  inertia  is  applicable  to  the  equation  of  forces. 

SI. 

B  = —  > 
e 

I          d3 

and  —  =  TTT-?»  (for  circular  section  of  diameter  d)i 

c        10.2  v 

Sd3 
therefore,  B  =  I0~2*  (32) 

For  a  hollow  shaft  or  pin  this  expression  becomes : 

B  = 
Combined  Stresses.     In  the  greater  number  of  cases  met  with 


104 


MACHINE  DESIGN 


in  practice,  we  find  two  or  more  simple  stresses  acting  at  the  same 
time,  and,  although  the  shaft  may  be  strong  enough  for  any  one  of 
them  alone,  it  may  fail  under  their  combined  action.  The  most 
common  cases  are  discussed  below. 

Tension  or  Pressure  Combined  with  Bending.  In  Fig.  28, 
the  load  W  produces  a  tension  acting  over  the  whole  area  of  d,  due 
to  its  direct  pull.  It  also  produces  a  bending  action  due  to  the 
leverage  K,  which  puts  the  fibers  at  B  in  tension  and  those  at  the 
opposite  side  in  compression.  It  is  evident,  therefore,  that  by 
taking  the  algebraic  sum  of  the  stresses  at  either  side  we  shall 
obtain  the  net  stress.  It  is  also  evident  that  the  greatest  and 


W 


Fig.  28. 


Fig.  29 


controlling  stress  will  occur  on  the  side  where  the  stresses  add,  or 
on  the  tension  side.     Hence,  from  mechanics, 


or, 


Also, 


4W 

S  =  — -p     (due  to  direct  tension).      (34) 

1T< '( 

Sd* 


or, 


10.2  WR 


(due  to  bending).       (35) 


Hence  the  combined  tensional  stress  acting  at  the  point  B,  or,  in 


MACHINE  DESIGN 


105 


fact,  at  any  point  on  the  extreme  outside  of  the  vertical   shaft  to- 
ward the  force  W,  is: 

4W      10.2  WR 

f-     -77T—  (36) 


If  W  acted  in  the  opposite  direction,  the  greatest  stress  would 
still  be  at  the  side  B,  but  would-be  a  compression  instead  of  a  ten- 
sion, of  the  same  magnitude  as  before. 

Tension  or  Compression  Combined  with  Torsion.  In  Fig.  29, 
Y  might  be  the  end  load  on  a  vertical  shaft;  and  the  two  forces  JV" 
might  act  in  conjunction  with  it  as  in  the  case  of  Fig.  26,  at  the 
radius  R.  This  case  is  not  very  often  met  with.  It  is  usually 
possible  to  combine  the  moments,  find  an  equivalent  moment  of  a 
simple  kind,  and  use  the  corresponding  simple  fiber  stress.  In  the 
case  in  question  we  have  a  direct  stress  to  be  combined  with  a 
shearing  stress,  and  mechanics  gives  us  the  following  solution : 


Fig.  30. 

Let  Ss  =  simple  shearing  stress  (Ibs.  per  sq  in.). 
Let  Sc  =  simple  compressive  stress  (Ibs.  per  sq.  in.). 
Let  Srs—  resultant  shearing  stress  (Ibs.  per  sq.  in.). 
Let  Src—  resultant  compressive  stress  (Ibs.  per  sq.  in.). 

We  then  have  : 


or, 
Also, 


=  5.1  ' 

6.1(2WR) 

3  ~          d* 


v  == 


(37) 


IT 


106  MACHINE  DESIGN 


Sc  =      rV.  (38) 

Now,  from  a  solution  given  in  simplest  form  in  u  Merriman's 
Mechanics"  —  which  the  student  may  consult,  if  desired  —  values 
for  the  resultant  stresses  can  be  found.  Whichever  of  these  is 
the  critical  one  for  the  material  used,  should  form  the  basis  for  its 
diameter: 


Srs  =  ^Ss*+i-  (39) 


Also,  Src=  =  -     +      Ss2+-  (40) 

Bending  Combined  with  Torsion.  In  Fig.  30,  the  load  W 
acts  not  only  to  twist  the  shaft  off,  but  also  presses  it  sidewise 
against  the  bearing.  As  it  is  usually  customary  to  figure  the 
maximum  moment  as  taking  place  at  the  center  of  the  bearing, 
the  length  L,  which  determines  the  bending  moment,  is  taken  to 
that  point.  The  theory  of  the  stress  induced  in  this  case  is  com- 
plicated. In  order  to  make  the  magnitude  of  the  moments  clearer, 
let  us  introduce  the  two  equal  and  opposite  forces  F  and  F1,  each 
equal  to  W,  at  the  point  C.  We  can  evidently  do  this  without 
changing  the  equilibrium  of  the  shaft  in  anyway.  We  now  see 
that  W  and  F1  act  as  a  couple  giving  a  twisting  moment  WR  ; 
and  that  F  acts  with  a  leverage  L,  producing  a  bending  moment 
FL  =  WL,  at  the  middle  of  the  bearing. 

If,  now,  we  find  an  equivalent  twisting  moment,  or  an  equiv- 
alent bending  moment,  which  would  produce  the  same  effect  on 
the  fibers  of  the  shaft  as  the  two  combined,  we  can  treat  the  cal- 
culation of  the  diameter  as  a  simple  case,  and  proceed  as  in  the 
cases  of  simple  torsion  and  simple  bending  considered  above.  This 
relation  is  given  us  in  mechanics: 


E 


-  E 


(42) 

These  expressions  are  true  in  relation  to  each  other,  on  the  assump- 
tion that  the  allowable  fiber  stress  S  is  the  same  for  tension,   com- 


MACHINE  DESIGN  107 


pression,  and  shearing.  For  the  material  of  which  shafts  are  usu- 
ally made,  this  is  near  enough  to  the  truth  to  give  safe  and  practi- 
cal results.  Using  the  expressions  for  internal  niomehts~~of~resist- 
ance  as  previously  noted  for  circular  sections,  we  then  have  : 


Also,  Te=|*. 

Either  equation  may  be  used  ;  the  diameter  d  will  result  the  same 
whichever  equation  is  taken.  For  the  sake  of  simplicity,  equation 
42  is  generally  preferred,  equation  44  being  taken  in  conjunction 
with  it. 


The  expression  V  B2  -f-  T2  is  one  that  would  be  a  long  and 
tedious  task  to  calculate.  By  inspection  it  is  readily  seen  that 
this  quantity  can  be  graphically  represented  by  means  of  a  right- 
angled  triangle  having  B  and  T  as  the  sides.  We  may  then  lay 
down  on  a  piece  of  paper,  to  some  convenient  scale,  the  moments 
B  and  T  as  the  sides  of  a  right-angled  triangle,  when,  upon 
measuring  the  hypothenuse,  we  can  easily  read  off  to  the  same 
scale  V  BJ  +  T2.  Even  if  the  drawing  is  made  to  a  small  scale, 
the  accuracy  of  the  reading  wTill  be  sufficient  to  enable  the  value 
for  d  to  be  solved  very  closely.  This  graphical  method  is  illus- 
trated in  Part  I. 

Deflection.  For  a  shaft  subjected  to  pure  torsion,  as  in  Fig. 
26,  the  angular  deflection  due  to  the  load  may  be  carried  to  a  cer- 
tain point  before  the  limit  of  working  fiber  stress  is  exceeded. 
The  equation  worked  out  from  mechanics  for  this  condition,  is: 


which  at  once  gives  the  number  of  degrees  of  angular  deflection 
for  a  shaft  whose  modulus  of  elasticity,  torsional  moment,  and 
length  are  known. 

The  shearing  modulus  of  elasticity  of  ordinary  shaft  steel  runs  from 
10,000,000  to  13,000,000,  giving  as  an  averare  about  12,000,000. 

By  the  well-known  relation  of  ''Hooke's  lawT  "  (stresses  pro- 

portional to  strains  within  the  elastic  limit  of  the  material),  we  have: 


108  MACHINE  DESIGN 


A°          SL 


360° 

,      x 
=  -  (46) 


A  twist  of  one  degree  in  a  length  of  twenty  diameters  is  a 
usual  allowance.  Substituting  A  =  1,  L  —  20d,  and  G  =  12,000, 
000,  we  have: 

S  =  5,240  (nearly).  (47) 

This  is  a  safe  value  for  shearing  fiber  stress  in  steel.  In  fact,  in 
calculations  for  strength,  even  for  reversing  stresses,  the  usual 
figure  is.  8,000  (Ibs.  per  square  inch),  thus  indicating  that  the  re- 
lation of  one  degree  to  twenty  diameters  is  well  within  the  limit 
of  strength. 

For  a  hollow  shaft  the  above  formula  becomes  : 

584  TL 

(48) 


Transverse  deflection  occurs  when  the  shaft  is  subjected  to  a 
bending  moment.  It  may  therefore  exist  alone  or  in  conjunction 
with  angular  deflection.  Transverse  deflection  of  shafts,  however, 
rarely  exists  up  to  the  point  of  limiting  fiber  stress,  because  before 
that  point  is  reached  the  alignment  of  the  shaft  is  so  disturbed 
that  it  is  not  practicable  as  a  device  for  transmitting  power.  A 
transverse  deflection  of  .01  inch  per  foot  of  length  is  a  common 
allowance  ;  but  it  is  impossible  to  fix  any  -general  limit,  as  in  many 
casss  this  figure,  if  exceeded,  would  do  no  harm,  while  in  others- 
such  as  heavily  loaded  or  high-speed  bearings — even  the  figure 
given  might  be  fatal  to  good  operation. 

The  formula  for  transverse  deflection,  deduced  from  mechan- 
ics, varies  with  the  system  of  loading.  The  three  most  common 
conditions  only  are  given  below,  reference  to  the  handbook  being 
necessary  if  other  conditions  must  be  satisfied: 

Fixed  at  one  end,  loaded  at  the  other, 

(49) 


OF  THE 

UNIVERSITY 


or 


MACHINE  DESIGN  100 


Supported  at  ends,  loaded  in  middle, 

WL3 


e  = 


48  El 

Supported  at  ends,  loaded  uniformly, 

5WL3 


(50) 


(51) 


For  transverse  deflection  the  direct  modulus  of  elasticity  must 
be  used,  for  the  fibers  are  stretched  or  compressed,  instead  of  being 
subjected  to  a  shearing  action.  The  most  usual  value  of  the  di- 
rect modulus  of  elasticity  for  ordinary  steel  is  30,000,000,  and  is 
denoted  in  most  books  by  the  symbol  E.  Both  the  shearing  and 
direct  moduli  of  elasticity  are  really  nothing  but  the  ratio  of  the 
stress  to  the  strain  produced  by  that  stress,  it  being  assumed  that 
the  given  material  is  perfectly  elastic.  A  material  is  supposed  to 
be  perfectly  elastic  up  to  a  certain  limit  of  stress,  and  it  is  within 
this  limit  that  the  relation  as  above  holds  good. 

o 

Expressed  in  the  form  of  an  equation  this  would  be  : 

•-1-4-       <*> 

L 

Centrifugal  Whirling.  If  a  line  shaft  deflect  but  slightly, 
due  to  its  own  weight,  or  the  weight  or  pressure  of  other  bodies 
upon  it,  and  then  be  run  at  a  high  speed,  the  centrifugal  force  set 
up  increases  the  deflection,  and  the  shaft  whirls  about  the  geomet- 
rical line  through  the  centers  of  the  bearings,  causing  vibration 
and  wear  in  the  adjoining  members.  It  is  evident  that  the  prac- 
tical remedy  for  this  tendency  in  a  shaft  of  given  diameter  and 
speed  is  to  locate  the  bearings  sufficiently  close  to  render  the  action 
of  small  effect. 

Many  formulae  might  be  given  for  this  relation,  each  being 
based  on  different  assumptions.  Perhaps  as  widely  applied  and 
as  simple  as  any,  is  the  "  Rankine  "  formula,  which  sets  the  limit 
of  length  between  bearings  for  shafts  not  greatly  loaded  by  inter- 
mediate pulleys  or  side  strains  : 


M  = 


110 


MACHINE  DESIGN 


Horse=Power  of  Shafting.  Horse-power  is  a  certain  specific 
rate  of  doing  work,  vis.,  33,000  foot-pounds  per  minute.  Hence, 
to  find  the  horse-power  that  a  shaft  will  transmit,  we  must  first 
find  the  work  done,  and  then  relate  it  to  the  speed.  Take,  for  ex- 
ample, the  case  of  a  pulley,  the  symbols  being  the  same  as  before 
—namely,  P  =  driving  force  at  rim  of  pulley  (Ibs.);  R  —  radius 
of  pulley  (inches) ;  N  =  number  of  revolutions  per  minute;  and 
II  =  horse- power.  Then, 

Work  =  force  X  distance  =  P  X  (2  TT  RN)  =  H  X  33,000  X  12; 


or, 


63,02511 
N 


(54) 


This  is  one  of  the  most  useful  equations  for  calculations  involving 
horse-power.  By  it  the  number  of  inch-pounds  torsion  for  any 
horse-power  can  be  at  once  ascertained. 

It  should  be  clearly  noted,  however,  that  in  this  equation  the 
bending  moment  does  not  enter  at  all.  Hence  any  shaft  based  in 
size  on  horse-power  alone,  is  based  on  torsional  moment  alone, 
bending  moment  being  entirely  neglected.  In  many  cases  the 
bending  moment  is  the  controlling  one  as  to  limiting  fiber  stress. 
Hence  empirical  shafting  formulae  depending  upon  the  horse- 
power relation  are  unsafe,  unless  it  is  definitely  known  just  what 
torsional  and  bending  moments  have  been  assumed. 

The  only  safe  way  to  figure  the  size  of  a  shaft  is  to  find 
accurately  what  torsional  moment  and  bending  moment  it  has  to 
sustain,  and  then  combine  them  according  to  equation  41  or  42 


MACHESTE  DESIGN  III 

introducing  the  el$!nent  of  speed  as  basis  for  assumption  of  a  high 
or  low  working  fiber  stress. 

PRACTICAL  MODIFICATION.  The  practical  methods  of 
handling  the  theoretical  shaft  equations  have  reference  to  the  fit  of 
the  shaft  within  the  several  pieces  upon  it.  The  running  fit  of  a 
shaft  in  a  bearing  is  usually  considered  to  be  so  loose  that  the  shaft 
could"  freely  deflect  to  the  center  of  the  bearing.  This  is  doubtless 
an  extreme  view  of  the  case,  but  it  is  the  only  safe  assumption. 
Hence  a  shaft  running  in  bearings  (see  Fig.  31)  is  supposed  to  be 
supported  at  the  centers  of  those  bearings,  and  its  theoretical 
strength  is  based  on  this  supposition. 

For  a  tight  or  driving  fit  upon  the  shaft,  a  safe  assumption  to 
make  is  that  there  is  looseness  enough  at  the  ends  of  the  fit  to  per- 
mit the  shaft  to  be  stressed  by  the  load  a  short  distance  within  the 
faces  of  the  hub,  say  from  ^  inch  to  1  inch.  For  example,  refer- 
ring to  Fig.  31,  suppose  Pj  to  be  the  transverse  load,  exerted 
through  a  hub  fast  upon  the  part  of  the  shaft  d3.  Taking  mo- 
ments about  the  center  of  one  bearing,  and  solving  for  the  reaction 
at  the  center  of  the  other,  we  have  : 

P1^=K1  K; 

or>  ^  =  -ir:  (55) 

Also,  P,  t  =  R2  K; 

*       R 


Now,  as  far  as  the  part  of  shaft  d3  is  concerned,  it  may  depend  for 
its  size  on  the  bending  moment  K2  J,  or  on  Rj  a.  The  reason  the 
lever  arm  is  not  taken  to  the  point  directly  under  the  load  P15  is 
because  it  is  not  practically  possible  to  break  the  shaft  at  that 
point,  on  account  of  the  reinforcement  of  the  hub,  which  is  tightly 
fitted  upon  it.  Trying  these  moments  to  see  which  is  the  greater, 
we  shall  find  that  the  greater  moment  always  occurs  in  connection 
with  the  longer  lever  arm.  Hence  K2  b  will  be  greater  than  Rj  a. 
We  then  write  the  equation  of  external  moment  =  internal  mo- 
ment: 

BJ-      8<V- 
K*  h  '     10.2    ' 


112  MACHINE  DESIGN 


d*  =  ^-^T  (57) 

For  the  size  of  bearing  A  we  have  the  maximum  bending  mo- 
ment: 

L,       S  d? 


For  the  size  of  bearing  B  we  have  the  maximum  moment: 


2  2  ""10.2 

3 


.        ,  ,      , 

<*:  v-nri  (59) 

The  above  calculations  are,  of  course,  on  the  assumption  that  no  torsion 
is  transmitted  either  way  through  this  axle.  We  should  in  that  case  have 
combined  torsion  and  bending.  This  has  been  made  sufficiently  clear  in  pre- 
ceding paragraphs  .and  in  Part  I,  to  require  no  further  illustration. 

The  dotted  line  in  Fig.  31  shows  the  theoretical  shape  the 
axle  should  take  under  the  assumed  conditions.  The  practical 
modification  of  this  shape  is  obvious.  At  the  shoulders  of  the 
shaft  the  corners  should  not  be  sharp,  but  carefully  filleted,  to 
avoid  the  possible  starting  of  a  crack  at  those  points. 

Often  the  diameter  of  certain  parts  of  a  shaft  may  be  larger 
than  strength  actually  calls  for.  For  example,  in  Fig.  31,  the 
part  d3  need  only  be  as  large  as  the  dotted  line;  but  it  is  obvious 
that  unless  the  key  is  sunk  in  the  body  of  the  shaft,  the  hub  could 
not  be  slipped  into  place  over  the  part  dt.  If,  however,  the  diam- 
eter d3  be  made  large  enough  so  that  the  bottom  of  the  key  will 
clear  d^  the  rotary  cutter  which  forms  the  key  way  in  d3  will  also 
clear  d^  and  the  key  way  can  be  more  easily  produced. 

In  cases  where  fits  are  not  required  to  be  snug,  a  straight 
shaft  of  cold-rolled  steel  is  commonly,  used.  Here  any  parts  fast- 
ened  on  the  middle  of  the  shaft  have  to  be  driven  over  a  consider- 
able length  of  the  shaft  before  they  reach  their  final  position. 
Moreover,  there  is  no  definite  shoulder  to  stop  against,  and  meas- 
urement has  to  be  resorted  to  in  locating  them. 


MACHINE  DESIGN  113 

It  does  not  pay  to  turn  any  portion  of  a  cold-rolled  shaft,  un- 
less it  be  the  very  ends,  for  relieving  the  "  skin  tension  "  in  such 
material  is  sure  to  throw  the  -  shaft  out  of  line  and  necessitate 
subsequent  straightening. 

Turned-steel  shafts  for  machines  may  with  advantage  be 
slightly  varied  in  diameter  wherever  the  fit  changes;  and  although 
the  production  of  shoulders  costs  something,  yet  it  assists  greatly 
in  bringing  the  parts  to  their  exact  location,  and  enables  the  work- 
man to  concentrate  his  best  skill  on  the  fine  bearing  fits,  and  to 
save  time  by  rough-turning  the  parts  that  have  no  fits. 

Hollow  shafts  are  practicable  only  for  large  sizes.  The  advan- 
tages of  removing  the  inner  core  of  metal,  aside  from  some  specific 
requirement  of  the  machine,  are  that  it  eliminates  all  possibility  of 
cracks  starting  from  the  checks  that  may  exist  at  the  center,  per- 
mits inspection  of  the  material  of  a  shaft,  and,  in  case  of  hollow- 
forged  shafts,  gives  an  opening  for  the  forging  mandrel.  In  the 
last  case,  the  material  is  improved  by  a  rolling  process. 

The  material  most  common  for  use  in  machine  shafting  is  the 
ordinary  "  Machinery  Steel,"  made  by  the  Bessemer  process.  This 
steel  is  apt  to  be  "seamy,"  and  often  contains  checks"  and  flaws 
that  are  detected  only  upon  sudden  and  unexpected  breakage  of  a 
part  apparently  sound.  This  characteristic  is  a  result  of  the  proc- 
ess employed  in  the  manufacture  of  the  steel,  and  thus  far  has 
never  been  wholly  eliminated.  Bessemer  steel  is,  nevertheless,  a 
very  useful  material,  and  the  above  weakness  is  not  so  serious  but 
that  this  kind  of  steel  can  be  used  with  success  in  the  great  majority 
of  cases. 

When  a  more  homogeneous  shaft  is  desired,  open=hearth  steel 
is  available.  This  is  a  more  reliable  material  to  use  than  the  Bes- 
semer, and  costs  somewhat  more.  It  makes  a  stiff,  true,  fine-sur- 
faced shaft,  high-grade  ^n  every  respect.  It  is  usually  specified 
for  armature  shafts  of  dynamos  and  motors. 

Steels  of  special  strength,  toughness,  and  elasticity  are  made 
under  numerous  processes.  Nickel  steel  is  perhaps  the  most  con- 
spicuous example.  While  for  this  steel  a  high  price  has  to  be 
paid,  yet  its  great  strength,  in  connection  with  other  valuable  qual- 
ities, makes  it  a  material  extremely  valuable  for  service  where  light 
weight  is  essential,  or  where  contracted  space  demands  small  size. 


114  MACHINE  DESIGN 

The  range  of  strength  of  these  various  steels  is  so  great  that  it  is  well- 
nigh  useless  to  go  into  a  discussion  of  it  here.  Keference  should  be  had  to 
the  extended  discussions  of  the  handbooks,  and  to  special  trade  pamphlets. 
A  study  of  the  possibilities  of  steel  in  its  various  forms  for  use  in  shafting, 
is  very  valuable  as  a  basis  for  design,  as  it  can  almost  be  said  that  a  machine 
consists  chiefly  of  a  ' '  collection  of  shafts  with  a  structure  built  round  them. ' ' 
The  shafts  are  like  a  core,  and  evidently  the  size  of  the  core  determines  the 
shell  about  it. 

PROBLEHS  ON  SHAFTS. 

1.  Required  the  twisting  moment  on  a  shaft  that  transmits 
30  horse-power  at  120  revolutions  per  minute. 

2.  Find  the  diameter  of  a  steel  shaft  designed  to  transmit  50 
horse-power  at  150  revolutions  per  minute. 

3.  Assuming  same  data  as  in  Problem  1,  find  the  diameters 
of  a  hollow  shaft  for  a  value  of  S  —  8,000. 

4.  A   belt   on   an   idler  pulley  embraces   an   angle   of   120 
degrees.     Assuming  tension   of  belt   17000  pounds  on  each  side, 
and  pulley  located  midway  between  bearings,  which  are  30  inches 
from  center  to  center,  what  is  the  diameter  of  shaft  required  ? 

5.  Calculate  the  diameter  of  a  steel  shaft  designed  to  transmit 
a   twisting  moment  of  400,000   inch- pounds  and  also  to  take  a 
bending  moment  of  300,000  inch-pounds. 

6.  Find  the  angular  deflection  in  a  4- inch  shaft  20  feet  long 
when  subjected  to  a  load   of   5,500  pounds  applied  to  an  arm  of 
30-inch  radius.     Assume  transverse  modulus  of  elasticity  equal  to 
12,000,000. 

7.  The   overhung  crank  of  a  steam  engine  has  a  force  of 
32,000  Ibs.  at  the  center  of  the  crank  pin,  which  is  12  inches  from 
the  center  of  the   shaft  bearing,  measured  parallel  to  the   shaft. 
The  radius  of  crank  arm  is  10  inches.     Assume  S  equal  to  10,000. 
Calculate  the  diameter  of  the  crank  shaft. 

8.  On  a  short,  vertical  steel  shaft  the  load  is  5,000  pounds. 
A  gear,  36  teeth,  1J  diametral  pitch,  at  top  of  shaft,  transmits  a 
load  of  4,000  pounds  at  the  pitch  line.     Safe  shear  =  7,500.     What 
is  the  diameter  of  the  shaft  ? 

SPUR  GEARS. 

NOTATION — The  following  notation,  is  used  throughout  the  chapter  on  Spur  Gears: 

b  = Breadth  of  rectangular  section  of       M,  Mi=Revolutions  per  minute. 

arm  (inches) .  ^11= Coefficient  of  friction  between  teeth. 


MACHINE  DESIGK  115 

C  = Width   of   arm    extended  to  pitch  N=Number  of  teeth, 

line  (inches).  n  =  Number  of  arms. 

c  =  Distance  from  neutral  axis  to  outer  P= Diametral  pitch  (teeth  per  inch  of 

fiber  (inches).  diameter). 

D=Pitch  diameter  of  gear  (inches).  Pi=Circular  pitch  (inches). 

F=Faceof  gear  (inches).  Q,  Qi= Normal  pressure  between  teeth 
f  =Clearance     of     tooth     at     bottom  (Ibs.). 

(inches).  R,  Ri= Resultant      pressure      between 
G= Thickness  of  arm  extended  to  pitch  teeth  (Ibs.). 

line  (inches).  r,  n= Radius  of  pitch  circles  (inches). 

H=Thickness  of  tooth  at  any  section  s  =Fiber   stress   of  material  (Ibs.  per 

(inches).  sq.  in.). 

h  =  Depth  of  rectangular  section  of  arm  s  =  Addendum    of    tooth  (inches) =De- 

(inches).  dendum  of  tooth. 

I  =  Moment  of  inertia.  t  ^Thickness   of   tooth   at   pitch   line 
K=Thickness  of  rim  (inches).  (inches). 

L=  Distance  from  top  of  tooth  to  any  W=Load  at  pitch  line  (Ibs.). 

section  (inches).  y  =coefflcient  for  "  Lewis  "  formula. 

ANALYSIS.  If  a  cylinder  be  placed  on  a  plane  surface,  with 
its  axis  parallel  to  the  plane,  an  attempt  to  rotate  the  cylinder 
about  its  axis  would  cause  it  to  roll  on  the  plane. 

Again,  if  two  cylinders  be  provided  with  axial  bearings,  and 
be  slightly  pressed  together,  motion  of  one  about  its  axis  will 
cause  a  similar  motion  of  the  other,  the  two  surfaces  rolling  one 
on  the  other  at  their  common  tangent  line.  If  moved  with  care, 
there  will  be  no  slipping  in  either  of  the  above  cases — which  is 
explained  by  the  fact  that  no  matter  how  smooth  the  surfaces  may 
appear  to  be,  there  is  still  sufficient  roughness  to  make  the  little 
irregularities  interlock  and  act  like  minute  teeth. 

The  magnitude  of  the  force  possible  to  be  transmitted  de- 
pends not  only  on  the  roughness  of  the  surfaces,  but  on  the 
amount  of  pressure  between  them.  Suppose  that  one  cylinder  is 
a  part  of  a  hoisting  drum,  on  which  is  wound  a  rope  with  a  weight 
attached.  We  can  readily  make  the  weight  so  great  that,  no  mat- 
ter how  hard  we  press  the  two  cylinders  together,  the  driving 
cylinder  will  not  turn  the  hoisting  cylinder,  but  will  slip  past  it. 
If  now,  instead  of  increasing  the  pressure,  which  is  detrimental 
both  to  cylinders  and  bearings  of  same,  we  increase  the  coarseness 
of  the  surfaces,  or,  in  other  words,  put  teeth  of  appreciable  size 
on  these  surfaces,  we  attain  the  desired  result  of  positively  driving 
without  excessive  side  pressure. 

These  artificial  projections,  or  teeth,  must  fit  into  one  another; 
hence  the  surfaces  of  the  original  cylinders,  having  been  broken 
up  into  alternate  projections  and  hollows,  have  entirely  disap- 


116 


MACHINE  DESIGN 


peared  to  the  eye;  they  nevertheless  exist  as  ideal  or  imaginary 
surfaces,  which  roll  together  with  the  same  surface  velocities  as  if 
in  bodily  form,  provided  that  the  curves  of  the  teeth  are  correctly 
formed.  Several  mathematical  curves  are  available  for  use  as 
tooth  outlines,  but  in  practice  the  involute  and  cycloidal  curves 
are  the  only  ones  used  for  tnis  purpose. 

The  ideal  surfaces  are  known  as  pitch  cylinders  or  pitch 
circles.  In  Fig.  32  is  shown  an  end  view  of  such  a  pair  of  cylin- 
ders in  contact  at  their  piteh  point  P.  In  gear  calculations  we 
assume  that  there  is  no  slip  between  the  pitch  circles,  acting  as 
driving  cylinders;  hence  the  speeds  of  the  two  pitch  circles  at  the 


Fig.  32. 


pitch  point  are  equal.     If  M  and  Mx  be  the  revolutions  per  minute 
of  the  cylinders  respectively,  r  and  r^  their  radii,  then 


or, 


M 


(60) 


That  is,  the  number  of  revolutions  varies  inversely  as  the  radii. 

The  simple  calculation  as  above  is  the  key  to  all  calculations 
involving  gear  trains  in  reference  to  their  speed  ratio. 

Fig.  33  represents  cycloidal  teeth  in  the  two  extreme  positions 
of  beginning  and  ending  contact.  The  normal  pressure  Q  or  Q1 
between  the  teeth  in  each  position  acts  through  the  pitch  point  O, 
as  it  must  always  do  in  order  to  insure  the  condition  of  ideal  roll- 


MACHINE  DESIGN 


117 


ing  of  the  pitch  circles,  and  the  velocity  ratio  proportional  to  - 

As  the  surfaces  of  the  teeth  slide  together,  frictional  resistance  is 
produced  at  their  point  of  contact.  This  force  is  widely  variable, 
depending  on  the  material  and  condi'tion  of  the  tooth  surfaces, 
whether  smooth  and  well  lubricated,  or  rough  and  gritty.  As  this 
resistance  acts  in  conjunction  with  the  normal  Wee  between  the 
teeth,  we  may  construct  a  parallelogram  of  forces  on  these  two  as 
a  base,  the  resultant  pressure  between  the  teeth  being  slightly 
changed  thereby,  as  shown  in  Fig  33. 

Assuming  a  coefficient  of  friction  fJL,  the  force  of  friction  is  [L  Q  or  \L  Qi 
and  the  resultant  pressure  R  or  RI. 

Tooth  B  of  the  FOLLOWER  is  therefore  under  a  heavy  bending  moment 
measured  by  the  product  RL,  L 
being  the  perpendicular  distance 
from  the  center  of  the  tooth  at 
its  base  to  the  line  of  the  force. 
This  tooth  also  has  a  relatively 
small  compressive  stress  due  to 
the  resolved  part  of  R  along  the 
radius,  and  a  relatively  small 
shearing  stress  due  to  the  re- 
solved part  of  R  along  a  tangent 
to  the  pitch  circle. 

Tooth  D  of  the  driven  wheel 
or  FOLLOWER  has  a  relatively 
large  shearing  stress,  a  small 
bending  moment,  and  practi- 
cally no  direct  compressive 
stress. 

Tooth  A  of  the  driving  wheel 
or  DRIVER  has  a  relatively  large 
shearing  stress,  a  small  bending  moment,  and  small  compressive  stress. 

Tooth  C  of  the  DRIVER  has  a  large  bending  moment,  but  small  com 
pressive  and  shearing  stresses. 

The  conditions  as  noted  above  are  not  those  of  every  pair  of 
gears,  in  fact  they  vary  with  every  difference  of  pitch  circle,  or  of 
detail  and  position  of  tooth.  It  is  true,  however,  that  in  nearly 
all  cases  in  practice  the  bending  stress  is  the  controlling  one  from 
a  theoretical  standpoint.  Moreover,  the  designer  must  consider 
the  form  and  strength  of  the  tooth  when  it  is  under  the  condition 
of  maximum  moment.  This  evidently,  from  the  above,  occurs  at 
the  beginning  of  contact,  for  the  follower  teeth;  and  at  the  end  of 
contact,  for  the  driver  teeth.  In  the  particular  case  illustrated  in 


118  MACHINE  DESIGN 


Fig.  33,  if  the  material  in  both  gears  were  the  same,  tooth  C, 
being  the  weaker  at  the  root,  would  probably  break  before  B;  but 
if  C  were  of  steel,  and  B  of  cast  iron,  B  might  break  first. 

It  will  be  noticed  that  R  is  nearly  parallel  to  the  top  of  the 
tooth;  and  it  may  easily  happen  that  the  friction  may  become  of 
such  a  value  that  it  will  turn  the  direction  of  E,  until  it  lies  along 
the  top  of  the  tooth  exactly,  which  is  the  condition  for  maximum 
moment.  For  strength  calculations  it  is  usual  to  consider  this 
condition  as  existing  in  all  cases. 

At  the  beginning  of  contact  there  is  more  or  less  shock  when 
the  teeth  strike  together,  and  this  effect  is  much  more  evident  at 
high  speeds.  There  is  also  at  the  beginning  of  contact  a  sort  of 
chattering  action  as  the  driving  tooth  rubs  along  the  driven  tooth. 

Uniform  distribution  of  pressure  along  the  face  of  the  tooth  is 
often  impaired  by  uneven  wear  of  the  bearings  supporting  the  gear 
shafts,  the  pressure  being  localized  on  one  corner  of  the  tooth.  The 
same  effect  is  caused  by  the  accidental  presence  of  foreign  material 
between  the  teeth.  Again,  in  cast  gearing,  the  spacing  may  be 
irregular,  or,  on  account  of  draft  on  the  pattern,  the  teeth  may  bear 
at  the  high  points  only.  While  it  is 
usual  to  consider  that  the  load  is  evenly 
distributed  along  the  face  of  the  tooth, 
yet  the  above  considerations  show  that 
an  ample  margin  of  strength  must  al- 
ways be  allowed  on  account  of  these 
uncertainties. 

When  the  number  of  teeth  in  the 
mating  gears  is  high,  the  load  will  be 
distributed  between  several  teeth  ;  but, 
•as  it  is  almost  certain  that  at  some  time  Fig.  34. 

the  proper  distribution  of  load  will  not 

exist,  and  that  one  tooth  will  receive  the  full  load,  it  is  considered 
that  practically  the  only  safe  method  is  so  to  design  the  teeth  that 
a  single  tooth  may  be  relied  upon  to  withstand  the  full  load  without 
failure. 

THEORY.  Based  on  the  Analysis  as  given,  the  theory  of  gear 
teeth  assumes  that  one  tooth  takes  the  whole  load,  and  that  this  load 
is  evenly  distributed  along  the  top  of  the  tooth  and  acts  parallel  with 


MACHINE  DESIGN  119 


?ts  base,  thus  reducing  the  condition  of  the  tooth  to  that  of  a 
cantilever  beam.  The  magnitude  of  this  load  at  the  top  of  the 
tooth  is  taken  for  convenience  the  same  as  the  force  transmitted  at 
the  pitch  circle.  This  condition  is  shown  in  Fig.  34.  Equating 
the  external  moment  to  the  internal  moment,  we  then  have,  from 
mechanics: 


.          .. 

The  thickness  H  is  usually  taken  either  at  the  pitch  line  or  at 
the  root  of  the  tooth  just  before  the  fillet  begins;  and  L,  of 
course,  is  dependent  on  the  tooth  dimensions.  The  formula  is 
most  readily  used  when  the  outline  of  the  tooth  is  either  assumed 
or  known,  a  trial  calculation  being  made  to  see  if  it  will  stand  the 
load,  and  a  series  of  subsequent  calculations  followed  out  in  the 
same  way  until  a  suitable  tooth  is  found.  This  method  is  pursued 
because  there  are  certain  even  pitches  which  it  is  desirable  to  use; 
and  it  is  safe  'tc  say  that  any  calculation  figured  the  reverse  way 
would  result  in  fractional  pitches.  The  latter  course  may  be  used, 
however,  and  the  nearest  even  pitch  chosen  as  the  proper  one. 

As  stated  under  "Analysis,"  there  are  a  great  many  circum- 
stances attending  the  operation  of  gears  which  make  impossible 
the  purely  theoretical  application  of  the  beam  formulae.  For  this 
reason  there  is  no  one  element  of  machinery  which  depends  so 
much  on  experience  and  judgment  for  correct  proportion  as  the 
tooth  of  a  gear.  Hence  it  is  true  that  a  rational  formula  based  on 
the  theoretical  one  is  really  of  the  greater  practical  value  in  tooth 
design. 

If  we  examine  formula  61,  we  find  that  in  a  form  solved  for 
W,.we  have: 

SFH2 


Of  these  quantities,  H  and  L  are  the  only  variables,  for  we  can 
make  the  others  what  we  choose.  H  and  L  depend  upon  the 
circular  pitch  P1  and  the  curvature  and  outline  of  the  tooth.  If 
now  we  could  settle  ok  a  standard  system  of  teeth,  we  could  estab- 
lish a  coefficient  to  be  used  to  take  the  place  of  the  variable  part 


120 


MACHINE  DESIGN 


of  H  and  L,  which  depends  on  the  outline  of  tooth,  and  we  should 
thus  have  an  empirical  formula  which  would  be  on  a  theoretical 
basis. 

This,   Mr.   "Wilfred  Lewis  has  done;    and  it  is    safe  to  say 
that  this  formula  is  more  universally  used  and  with  more  satis- 


Fig.  35. 

factory  practical  results  than,  any  other  formula,  theoretical  or 
practical,  that  has  ever  been  devised.  His  coefficient  is  known  as 
T/,  and  was  determined  from  many  actual  drawings  of  different 
forms  of  teeth  showing  the  weakest  section.  .  This  coefficient  is 
worked  out  for  the  three  most  common  systems  as  follows : 


For  20°  involute,     y  =  0.154  - 
=  0.124 


8 

0.684 


For  15°  involute 
and  cycloidal, 


0  27fi 
For  radial  flanks,  y  ==  0.075  -         '    . 


(63) 

(64) 
(65) 


The  tooth  upon  which  the  above  is  based  is  the  American  standard  or 
Brown  &  Sharpe  tooth,  for  which  the  proportions  are  shown  in  Fig.  35. 

The  "  Lewis  "  formula*  is: 

W  =  SP1  Fy.  (66) 

A  table  indicating  the  value  of  S  for  different  speeds  follows: 

Safe  Working  Stresses  for  Different  Speeds. 


Speed  of  teeth, 
ft.  per  min. 

100 

200 

300 

600 

900 

1200 

1800 

2400 

Cast  iron 

8000 

6000 

4800 

4000 

3003 

2400 

2000 

170G 

Steel 

20000 

15000 

12000 

10000 

7500 

6000 

5000 

4300 

*NOTE.     A  full  and  convenient  statement  of  the  Lewis  formula   will 
be  found  in  "Kent 's  Pocket  Book.  " 


MACHIKE  DESIGK 


121 


A  usual  relation  of  F  to  Pl  is: 


For  cast  teeth,  F  =  2P1  to  3P1, 
For  cut  teeth,  F  -  3P1  to  4P1. 


(66) 

(67) 


The  usual  method  of  handling  these  formulae  is  as  follows: 

The  pitch  circles  of  the  proposed  gears  are  known  or  can  be  assumed; 
hence  W  can  readily  be  figured,  also  the  speed  of  the  teeth,  whence  S  can 
be  read  from  the  table.  The  desired  relation  of  F  to  P1  can  be  arbitrarily 
chosen,  when  P*  and  y  become  the  only  unknown  quantities  in  the  equation. 
A  shrewd  guess  can  be  made  for  the  number  of  teeth,  and  y  calculated  there- 
from. Then  solve  the  equation  for  P1  which  will  undoubtedly  be  fractional. 
Choose  the  nearest  even  pitch,  or,  if  it  is  desired  to  keep  an  even  diametral 
pitch,  the  fractional  pitch  that  will  bring  an  even  diametral  pitch.  Now, 
from  this  final  and  corrected  pitch,  and  the  diameter  of  the  pitch  circle, 
calculate  the  number  of  teeth  N  in  the  gear.  Check  the  assumed  value  of  y 
by  this  positive  value  of  N. 

Another  good  way  of  using  this  formula  is  to  start  with  the 
pitch  and  face  desired,  and  the  diameter  of  the  pitch  circle.     In 


Fig.  37. 

this  case  "W  is  the  only  unknown  quantity,  and  when  found  can  be 
compared  with  the  load  required  to  be  carried.  If  too  small, 
make  another  and  successive  calculations  until  the  result  approxi- 
mates the  required  load. 

SPUR  GEAR  Rin,  ARflS,  AND  HUB. 

ANALYSIS  and  THEORY.  The  rim  of  a  gear  has  to  transmit 
the  load  on  the  teeth  to  the  arms.  It  is  thus  in  tension  on  one  side 
of  the  teeth  in  action,  and  in  compression  on  the  other.  The  sec- 
tion of  the  rim,  however,  is  so  dependent  on  other  practical  con- 
siderations which  call  for  an  excess  of  strength  in  this  respect,  that 


122 


MACHINE  DESIGN 


it  is  not  considered  worth  while  to  attempt  a  calculation  on  this 
basis. 

Gears  seldom  run  fast  enough  to  make  necessary  a  calculation 
for  centrifugal  force  ;  and  in  general  it  can  be  said  that  the  design 
of  the  rim  is  entirely  dependent  on  practical  considerations.  These 
will  appear  later  under  "  Practical  Modification.  " 

The  arms  of  a  gear  are  stressed  the  same  as  pulley  arms,  the 
same  theory  answering  for  both,  except  that  a  gear  rim  always  be- 
ing much  heavier  than  a  pulley  rim,  the  distribution  of  load 
amongst  the  arms  is  better  in  the  case  of  a  gear  than  of  a  pulley, 
and  it  is  usually  safe  to  assume  that  each  arm  of  a  gear  takes  its  full 
proportion  of  load  ;  or,  for  an  oval  section,  equating  the  external 
moment  to  the  internal  moment  as  in  the  case  of  pulleys,  we  have  : 


WD 

n2 


=  0.0393  SA3. 


(68) 


Heavy  spur  gears  have  the  arms  of  a  cross  or  T  section  (Fig. 


* 

L 


Fig.  38. 

37),  the  latter  being  especially  applicable  to  the  case  of  bevel  gears 
where  there  is  considerable  side  thrust.  The  simplest  way  of 
treating  such  sections  is  to  consider  that  the  whole  bending  moment 
is  taken  by  the  rectangular  section  whose  greater  dimension  is  in 
the  direction  of  the  load.  The  rest  of  the  section,  being  close  to 
the  neutral  axis  of  the  section,  is  of  little  value  in  resisting  the 
direct  load,  its  function  beingf  to  give  sidewise  stiffness.  The 

O  D 

equation  for  the  cross  or  T  style  of  arm,  then  is  : 
W        D        SM2 

(6p) 


it 


6 


MACHINE  DESIGN 


Either  b  or  h  may  be  assumed,  and  the  other  determined.  As  a 
guide  to  the  section,  b  may  be  taken  at  about  the  thickness  of  the 
tooth. 

Gear  hubs  are  in  no  wise  different  from  the  hubs  of  pulleys  or 
other  rotating  pieces.  The  depth  necessary  for  providing  suffi- 
cient strength  over  the  key  to  avoid  splitting  is  the  guiding  ele- 
ment, and  can  usually  be  best  determined  by  careful  judgment. 

PRACTICAL  MODIFICATION.  The  practical  requirements, 
which  no  theory  will  satisfy,  are  many  and  varied.  Sudden  and 
severe  shock,  excessive  wear  due  to  an  atmosphere  of  grit  and  corros- 
ive elements,  abrupt  reversal  of  the  mechanism,  the  throwing-in  of 
clutches  and  pawls,  the  action  of  brakes  —  these  and  many  other 
influences  have  an  important  bearing  on  gear  design,  but  not  one 
that  can  be  calculated.  The  only  method  of  procedure  in  such 
cases  is  to  base  the  design  on  analysis  and  theory  as  previously 
given,  and  then  add  to  the  face  of  gear,  thickness  of  tooth,  or  pitch 
an  amount  which  judgment  and  experience  dictate  as  sufficient. 

Excessive  noise  and  vibration  are  difficult  to  prevent  at  high 
speeds.  At  1,000  feet  per  minute,  gears  are  apt  to  run  with  an 
unpleasant  amount  of  noise.  At  speeds  beyond  this,  it  is  often 
necessary  to  provide  mortise  teeth,  or  teeth  of  hard  wood  set  into 
a  cast-iron  rirn  (see  Fig.  38).  Rawhide  pinions  are  useful  in  this 
regard.  Fine  pitches  with  a  long  face  of  tooth  run  much  more 
smoothly  at  high  speeds  than  a  coarse  pitch  and  narrow-faced  tooth 
of  equal  strength.  Greater  care  in  alignment  of  shafts,  however, 
is  necessary,  also  stiffer  supports. 

Should  it  be  impracticable  to  use  a  standard  tooth  of  sufficient 
strength,  there  are  several  ways  in  which  we  can  increase  the 
carrying  capacity  without  increasing  the  pitch.  These  are: 

1.  Use  a  stronger  material,  such  as  steel. 

2.  Shroud  the  teeth. 

3.  Use  a  hook  tooth. 

4.  Use  a  stub  tooth. 

Shrouding  a  tooth  consists  in  connecting  the  ends  of  the  teeth 
with  a  rim  of  metal.  When  this  rim  is  extended  to  the  top  of  the 
tooth,  the  process  is  called  "  full  -shrouding"  (Fig.  39);  and  when 
carried  only  to  the  pitch  line,  it  is  termed  "  half  -shrouding  " 
(Fig.  40).  The  theoretical  effect  of  shrouding  is  to  make  the  tooth 


124 


MACH1KE  DESIGK 


act  -  like  a  short  beam  built  in  at  the  sides;  and  the  tooth  will 
practically  have  to  be  sheared  out  in  order  to  fail.  This  modifica- 
tion of  gear  design  requires  the  teeth  to  be  cast,  as  the  cutter 
cannot  pass  through  the  shrouding.  The  strength  of  the  shrouded 
gear  is  estimated  to  be  from  25  to  50  per  cent  above  that  of  the 
plain-tooth  type. 


Fig.  39. 


Fig.  40. 


The  hook-tooth  gear  (Fig.  41)  is  applicable  only  to  cases 
where  the  load  on  the  tooth  does  not  reverse.  The  working  side 
of  the  tooth  is  made  of  the  usual  standard  curve,  while  the  back  is 
made  of  a  curve  of  greater  obliquity,  resulting  in  a  considerable 
increase  of  thickness  at  the  root  of  the  tooth.  A  comparison  of 
strength  between  this  form  and  the  standard  may  be  made  by 
drawing  the  two  teeth  for  a  given  pitch,  measuring  their  thickness 
just  at  top  of  the  fillet,  and  finding  the  relation  of  the  squares 
of  these  dimensions.  The  truth  of  this  relation  is  readily  seen  from 
an  inspection  of  formula  61. 

The  stub  tooth  merely  involves  the  shortening  of  the  height 


MACHINE  DESIGN  125 

of  the  tooth  in  order  to  reduce  the  lever  arm  on  which  the  load 
acts,  thus  reducing  the  moment,  and  thereby  permitting  a  greater 
load  to  be  carried  for  the  same  stress. 

The  rim  of  a  gear  is  dependent  for  its  proportions  chiefly  on 
questions  of  practical  moulding  and  machining.  It  must  bear  a 
certain  relation  to  the  teeth  and  arms,  so  that,  when  it  is  cooling  in 
the  mould,  serious  shrinkage  stresses  will  not  be  set  up,  forming 
pockets  and  cracks.  Moreover,  when  under  pressure  of  the  cutter 
in  the  producing  of  the  teeth,  it  must  not  chatter  or  spring.  This 
condition  is  quite  well  attained  in  ordinary  gears  when  the  thick- 
ness of  the  rim  below  the  base  of  the  tooth  is  made  about  the  same 
as  the  thickness  of  the  tooth. 


UGHT  PRESSURE: 
ON  BACK  OF  TOOTH. 

35°  INVOLUTE 


LOADED  SIDE 
INVOLUTE. 


Fig.  41. 

The  stiffening  ribs  and  arms  must  all  be  joined  to  the  rim  by 
ample  fillets,  and  the  cross-section  must  be  as  uniform  as  possible, 
to  prevent  unequal  cooling  and  consequent  pulling-away  of  the 
arms  from  the  rim  or  hub.  Often  the  calculated  size  of  the  arms 
at  both  rim  and  hub  has  to  be  modified  considerably  to  meet  this 
requirement. 

The  arms  are  usually  tapered  to  suit  the  designer's  eye,  a 
small  gear  requiring  more  taper  per  foot  than  a  large  one.  Both 
rim  and  hub  should  be  tapered  ^  inch  per  foot  to  permit  easy 
drawing-out  from  the  mould. 

The  proportions  given  in  the  following  table  have  been  used 
with  success  as  a  basis  of  gear  design  in  manufacturing  practice. 
The  table  will  serve  as  an  excellent  guide  in  laying  out,  and  can  be 
closely  followed,  in  most  cases  with  but  slight  modification. 
Web  gears  are  introduced  for  small  diameters  where  the  arms  begin 
to  look  awkward  and  clumsy. 


MACHINE  DESIGN 


Gear  Design  Data. 

Measurements  given  in  inches.    Letters  refer  to  Fig.  42. 


Diametral  pitch  .  . 

P 

1* 

It 

2 

2i 

3 

3J 

4 

5 

6 

8 

Face  

F 

61 

54- 

4| 

SJ 

31 

2J 

21 

2i 

IJr 

U 

Thickness  of  arm 
when  extended 
to  pitch  line  .... 

G 

if 

11 

ji 

1 

i 

if 

i 

li 

1 

| 

Width  of  arm  when 
extended    to 
pitch  line  ...  ... 

C 

4. 

31 

3 

21 

2i 

2 

13 

11 

13 

li 

«-»2 

^2 

^4 

X5 

^1 

-"•8 

if 

Thickness  of  rim  .  . 

K 

21 

2s 
§ 

2| 

i| 

li 

if 

il 

1 

I 

i 

Depth  of  rib  

1 

2 

15 

H 

H 

1 

2 

j 

i 

+ 

1 

Thickness  of  web. 

T 

1* 

1 

7 
8 

l 

1 

A 

i 

rV 

1 

A 

Number  of  arms,  6. 

Give  inside  of  rims  and  hub  a  draft  of  J  inch  per  foot. 

BEVEL  GEARS. 

NOTATION — The  following  notation  is  used  throughout  the  chapter  on  Bevel  Gears : 

O  D=Outside  diameter  (inches). 

P      =Diametral  pitch  related  to  pitch 

diameter  (teeth  per  inch). 
P1    =Circular  pitch    measured  on   the 

circumference  of  D  (inches). 
S      =  Working  strength  of  material  (Ibs. 

per  sq.  in.). 
s       =  Addendum,   or   height   of    tooth 

above  pitch  line  (inches). 
=  Depth  of  tooth  below  pitch  line 

(inches). 

=  Angle  of  top  of  tooth  (degrees). 
=Thickness  of  tooth   at  pitch  line 

(inches). 

Working  load  at  pitch  line  (Ibs.). 


A  =Apex  distance  at  pitch  element  of 

cone  (inches). 
Ai=Apex   distance   at  bottom   element 

of  tooth  (inches). 

B  =  Angle  of  bottom  of  tooth  (degrees). 
C  =  Pitch  angle  (degrees). 
D  =Pitch  diameter  (inches). 
E  =Radius  increment  of  gear  (inches). 
F  =Face  of  gear  (inches). 
S   =Clearance  at  bottom  (inches). 
G  =  Angle  of  face  (degrees). 
H  =  Cutting  angle  (degrees). 
~K.  =Radius       increment        of      pinion 

(inches). 

N  =  Number  of  teeth. 
Ni=Formative     number    of    teeth,    or 

the  number  corresponding  to  the 

spur  gear  on  which  the  outline  of 

tooth  is  made. 


W 

y      =Factor  in  "Lewis"  formula. 


ANALYSIS.     It  is  possible  to  consider  bevel  gears  as  the 

o 

general  case  of  which  spur  gears  are  a  special  form.     The  pitch 


MACHINE  DESIGN 


surfaces  of  spur  gears  described  above  as  cylinders,  mathematically 
considered,  are  cones  whose  vertices  are  infinitely  distant,  while 
bevel  gears  likewise  are  based  on  pitch  cones,  but  with  ^vertex  at 
some  finite  point,  common  to  the  mating  pair.  Hence,  as  we 
might  expect,  the  laws  of  tooth  action  are  similar  in  bevel  gears 
to  those  in  the  case  of  spur  gears.  The  profile  of  the  tooth  in  the 
former  case,  however,  is  based,  not  on  the  real  radius  of  the  pitch 
cone,  but  on  the  radius  of  the  normal  cone  ;  and  in  the  develop- 
ment of  the  outline  the  latter  is  treated  just  as  though  it  were  the 
radius  of  a  spur  gear.  The  tooth  thus  formed  is  wrapped  back  up- 
on the  normal  cone  face,  and  becomes  the  large  end  of  the  taper- 
ing bevel-gear  tooth  (see  Fig.  44). 


Fig.  42. 

The  teeth  of  bevel  gears,  being  simply  projections  with  bases  on 
the  pitch  cones,  have  a  varying  cross-section  decreasing  toward  the 
vertex  ;  also  a  trapezoidal  section  of  root,  the  latter  section  acting 
as  a  beam  section  to  resist  the  cantilever  moment  due  to  the  tooth 
load. 

The  arms  must,  as  in  the  case  of  spur  gears,  transmit  the  load 
from  the  tooth  to  the  shaft;  in  addition,  the  arms  of  a' bevel  gear 
are  subjected  to  a  side  thrust  due  to  the  wedging  action  of  the 
cones.  Hence  sidewise  stiffness  of  the  arms  is  more  essential  in 
this  type  of  gear  than  in  the  case  of  the  spur  gear. 

THEORY.  It  is  evident  that  the  calculation  of  tooth  strength 
based  on  a  trapezoidal  section  of  root  would  be  somewhat  compli- 


128 


MACHINE  DESIGN 


cated  ;  also  that  the  trapezoid  in  most  cases  would  be  but  little 
different  from  a  true  rectangle.  Hence  the  error  will  be  but 
slight  if  the  average  cross -section  of  the  tooth  be  taken  to  repre- 
sent its  strength,  and  the  calculation  made  accordingly. 


bo 


130  MACHINE  DESIGN 

Fig.  45  shows  a  bevel-gear  tooth  with  the  average  cross-sec- 
tion in  dotted  lines.  For  the  purpose  of  calculation,  the  assump- 
tion is  made  that  the  section  A  is  carried  the  full  length  of  the 
face  of  the  gear,  and  that  the  load  which  this  average  tooth  must 
carry  is  the  calculated  load  at  the  pitch  line  of  section  A.  This 
is  equivalent  to  saying  that  the  strength  of  a  bevel-gear  tooth  is 
equal  to  that  of  a  spur-gear  tooth  which  has  the  same  face,  and  a 
section  identical  with  that  cut  out  by  a  plane  at  the  middle  of  the 
bevel  tooth.  The'load,  as  in  the  case  of  the  spur  gear,  should  be 
taken  at  the  top  of  the  tooth;  and  its  magnitude  can  be  con- 
veniently calculated  at  the  mean  pitch  radius  of  the  bevel  face, 
without  appreciable  error. 

This  similarity  to  spur  gears  being  borne 
in  mind,  the  calculation  for  strength  needs  no 
further  treatment.  Once  the  average  tooth  is 
assumed  or  found  by  layout,  a  strict  following- 
out  of  the  methods  pursued  for  spur-gear 
teeth  will  bring  consistent  results. 

The  detail  design  of  a  pair  of  bevel  gears 
involves   some  trigonometrical  computations 
in  order  properly  to  dimension  the  drawing 
for  use  in   finishing  the  blanks  and   subse- 
Fig.  45.  quently  in  cutting  the  teeth,  or,  in  the  case 

of  cast  gears,  in  making  the  pattern.     These 

calculations,  although  simple,  are  yet  apt  to  be  tedious;  and  inac- 
curacies are  likely  to  creep  in  if  a  definite  system  of  relations 
be  not  maintained.  Hence  the  results  of  these  calculations  are 
given  below  in  condensed  and  reduced  form.  The  deduction  of 
these  formulae  is  a  simple  and  interesting  exercise  in  trigonometry; 
and  it  is  urged  that  they  be  worked  out  by  the  student  from  the 
figure,  in  which  case  he  will  feel  greater  confidence  in  their  use. 

Axes  of  Gears  at  90  Degrees. 

Use  subscript  1  for  gear;  P  for  pinion.    Letters  refer  to  Fig.  44. 


MACHINE  DESIGN  131 


f__L     ZL    JL  (73) 

7  ~  10  -  20  -  20P' 

tan  Cp  =  ^-;  tan  Ci  =  ^-  ~  474) 

s         2  sin  C  •.  ..                              /-., 

tanT   --£  =  — jq- (75) 

2.314  sin  C 


s  -f  /  =  A  tan  B  =  -  0.368P1.  (77) 

A  =  ZPrinC  =  5pVNi«  +  Np*  ^4-*/D*2  +  DP2-  (78> 

A1 A_  _  N  7 

~  cos  B  —  2P  cos  B  sin  C 

Gi  =  90°-(Ci  +  T);  Gp  =  90°  -  (Cp  +  T).  (80) 

E  =  S  cos  Ci  =  S  sin  Cp  .  (81 ) 

K  =  S  cos  Cp  =  S  sin  Ci.  (82) 

PRACTICAL  MODIFICATION.  The  practical  requirements 
to  be  met  in  transmission  of  power  by  bevel  gears  are  the  same  as 
for  spur  gears;  but  in  the  case  of  bevel  gears  even  greater  care  is 
necessary  to  provide  stiffness,  strength,  true  alignment,  and  rigid 
supports.  As  far  as  the  gears  themselves  are  concerned,  a  long 
face  is  desirable;  but  it  is  much  more  difficult  to  gain  the  ad- 
vantage of  its  strength  than  in  the  case  of  spur  gears,  because  full 
bearing  along  the  length  of  the  tooth  is  hard  to  guarantee. 

The  rim  usually  requires  a  series  of  ribs  running  to  the  hub 
to  give  required  stiffness  and  strength  against  the  side  thrust  which 
is  always  present' in  a  pair  of  bevel  gears.  Instead  of  arms,  the 
tendency  of  bevel-gear  design,  except  for  very  large  gears,  is  toward 
a  web  on  account  of  the  better  and  more  uniform  connection 
thereby  secured  between  rim  and  hub.  This  web  may  be  lightened 
by  a  number  .-of  holes,  so  that  the  resultant  effect  is  that  of  a  num- 
ber of  wide  and  flat  arms. 

The  hubs  naturally  have  to  be  fully  as  long  as  those  of  spur 
gears,  because  there  is  greater  tendency  to  rock  on  the  shaft,  due 
to  the  side  thrust  from  the  teeth,  mentioned  above. 

The  teeth  on  small  gears  are  cut  with  rotary  cutters,  at  least 
two  finishing  cuts  being  necessary,  one  for  each  side  of  the  taper- 
ing tooth.  The  more  accurate  method  is  to  plane  the  teeth  on  a 
special  gear  planer,  and  this  method  is  followed  on  all  gears  of 
any  considerable  size.  The  practical  requirement  here  is  that  no 
portion  of  the  hub  shall  project  so  as  to  interfere  with  the  stroke 


132  MACHINE  DESIGN 

of  the  planer  tool.  The  requirements  of  gear  planers  vary  some- 
what in  this  regard. 

Finally,  .after  all  that  is  possible  has  been  done  in  the  design 
of  the  gear  itself  to  render  it  suitable  to  withstand  the  varied 
stresses,  especial  attention  must  be  paid  to  the  rigidity  of  the 
supporting  shafts  and  bearings.  Bearings  should  always  be  close 
up  to  the  hubs  of  the  gears,  and,  if  possible  the  bearing  for  both 
pinion  and  gear  should  be  cast  in  the  same  piece.  If  this  is  not 
done,  the  tendency  of  the  separate  bearings  to  get  out  of  line  and 
destroy  the  full  bearing  of  the  teeth  is  difficult  to  control.  Thrust 
washers  are  desirable  against  the  hubs  of  both  pinion  and  gear; 
also  proper  means  of  well  lubricating  the  same. 

With  these  considerations  carefully  met,  bevel  gears  are  not 
the  bugbear  of  machine  design  that  they  are  sometimes  claimed 
to  be.  The  common  reason  why  bevel  gears  cut  and  fail  to  work 
smoothly,  is  that  the  gears  and  supports  are  not  designed  carefully 
enough  in  relation  to  each  other.  This  is  also  true  of  spur  gears, 
but  the  bevel  gear  will  reveal  imperfections  in  its  design  far  the 
more  quickly  of  the  two. 

WORM  AND  WORM  GEAR. 

NOTATION— The  following  notation  is  used  throughout  the  chapter  on  Worm  and 
Worm  Gear : 

D   =Pitch  diameter  of  gear  (inches).  PI  =Circular     pitch  =  Pitch  '  of   worm 
E   =  Efficiency  between  worm  shaft  and  thread  (inches). 

gear  shaft  (per  cent).  R   =Radius  of  pitch  circle  of  worm  gear 
f     =  Clearance     of    tooth     at     bottom  (inches). 

(inches).  s     =  Addendum  of  tooth  (inches). 

i     =  Index  of  worm  thread  (1  for  single'  T    =Twisting   moment    on   gear   shaft 

2  for  double,  etc.).  (inch-lbs.). 

L    =  Lead  of  worm  thread  (inches).  Tw=Twisting  moment  on  worm    shaft 
M   =  Revolutions    of     gear    shaft     per  (inch-lbs.). 

minute.  t    =  Thickness  of  tooth    at   pitch   line 
Mw= Revolutions    of   worm    shaft     per  (inches). 

minute.  W  =Load  at  pitch  line  (Ibs.). 
N   = Number  of  teeth  in  gear. 

ANALYSIS.  The  simplest  way  of  analyzing  the  case  of  the 
worm  and  worm  gear  is  to  base  it  upon  an  ordinary  screw 
and  nut.  Take,  for  example,  the  lead  screw  of  a  common  lathe. 
The  carriage  carries  a  nut,  through  which  the  lead  screw  passes. 
By  the  rotation  of  the  screw,  the  carnage,  being  constrained  by  the 
guides  to  travel  lengthwise  of  the  ways,  is  moved.  This  motion 


MACHINE  DESIGN  133 

4V 


is,  for  a  single-threaded  screw,  a  distance  per  revolution  equal  to 
the  lead  of  the  screw. 

Now,  suppose  that  the  carriage,  instead  of  sliding-along  the 
ways,  is  compelled  to  turn  about  an  axis  at  some  point  below  the 
ways.  Also,  suppose  the  top  of  the  nut  to  be  cut  off,  and  its  length 
made  endless  by  wrapping  it  around  a  circle  struck  from  the  center, 
about  which  the  carriage  rotates.  This  reduces  the  nut  to  a 
peculiar  kind  of  spur  gear,  the  partial  threads  of  the  nut  now 
having  the  appearance  of  twisted  teeth. 

This  special  form  of  spur  gear*  based  OR  tne  idea  of  a  threaded 
nut,  is  known  as  a  worm  gear,  and  the  screw  is  termed  a  worm. 
The  teeth  are  loaded  similarly  to  those  of  a  spur  gear,  but  with  the 
additional  feature  of  a  large  amount  of  sliding  along  the  tooth 
surfaces.  This,  of  course,  means  considerable  friction;  and  it  is  in 
fact  possible  to  utilize  the  worm  and  worm  gear  as  an  efficient 
device,  only  by  running  the  teeth  constantly  in  a  bath  of  oil. 
Even  then  the  pressures  have  to  be  kept  well  down  to  insure  the 
required  term  of  life  of  the  tooth  surfaces. 

It  is  evident  that  for  one  revolution  of  a  single-threaded  worm, 
one  tooth  of  the  gear  will  be  passed.  The  speed  ratio  between  the 
worm  gear  and  worm  shaft  will  then  be  equal  to  the  number  of 
teeth  in  the  gear,  which  is  relatively  great.  Hence  the  worm  arid 
worm  gear  are  principally  useful  in  giving  large  speed  reduction 
in  a  small  amount  of  space. 

THEORY.  The  theory  of  worm-wheel  teeth  is  complicated 
and  obscure.  The  production  of  the  teeth  is  simple,  a  dummy  worm 
with  cutting  edges,  called  a  "hob,"  being  allowed  to  carve  its  way 
into  the  worm-gear  blank,  thus  producing  the  teeth  and  at  the 
same  time  driving  the  worm  gear  about  its  axis. 

It  is  clear  that  if  we  know  the  torsional  moment  on  the  worm- 
gear  shaft,  and  the  pitch  radius  of  the  worm  gear,  we  can  find  the 
load  on  the  teeth  at  the  pitch  line  by  dividing  the  former  by  the 
latter.  Expressed  as  an  equation: 

WR  =  T;orW=  =  ^-.  (83) 

How  we  shall  consider  this  value  of  W  as  distributed  on  the 
teeth,  is  a  question  difficult  to  answer.  The  teeth  not  only  are 


134  ..        MACHINE  DESIGN 

curved  to  embrace  the  worm,  but  are  twisted  across  the  face  of  the 
gear,  so  that  it  would  be  practically  impossible  to  devise  a  purely 
theoretical  method  of  exact  calculation.  The  most  reasonable  thing 
to  do  is  to  assume  the  teeth  as  being  equally  as  strong  as  spur-gear 
teeth  of  the  same  circular  pitch,  and  to  figure  them  accordingly. 
It  is  probably  true,  however,  that  the  load  is  carried  by  more  than 
one  tooth,  especially  in  a  hobbed  wheel;  so  we  shall  be  safe  in 
assuming  that  two  —  and,  in  case  of  large  wheels,  three  —  teeth 
divide  the  load  between  them.  With  these  considerations  borne 
in  mind,  the  case  reduces  itself  to  that  of  a  simple  spur-gear 
tooth  calculation,  which  has  already  been  explained  under  the 
heading  "Spur  Gears." 

The  worm  teeth,  or  threads,  are  probably  always  stronger  than 
the  worm-gear  teeth;  so  no  calculation  for  their  strength  need  be 
made. 

The  twisting  moment  on  the  worm  shaft  is  not  determined  so 
directly  as  in  the  case  of  spur  gears.  The  relative  number  of 
revolutions  of  the  two  shafts  depends  upon  the  u  lead  "  of  the 
worm  thread  and  the  number  of  teeth  in  the  gear. 

Lead  (L)  is  the  distance  parallel  to  the  axis  of  the  worm  which 
any  point  in  the  thread  advances  in  one  revolution  of  the  worm. 
Pitch  (P1)  is  the  distance  parallel  to  the  axis  of  the  worm  between 
corresponding  points  on  adjacent  threads.  The  distinction  between 
lead  and  pitch  should  be  carefully  observed,  as  the  two  are  often 
confounded,  one  with  the  other. 

The  thread  may  be  single,  double,  triple,  etc.,  the  index  of  the 
thread  i,  being  1,  2,  3,  etc.,  in  accordance  therewith.  The  relation 
between  lead  and  pitch  may  then  be  expressed  by  an  equation,  thus: 

L  =  *P.     .        ;  (84) 

When  the  index  of  the  thread  is  changed  the  speed  ratio  is 
changed,  the  relation  being  shown  by  the  equation: 


If  the  efficiency  were  100  per  cent  between  the  two  shafts, 
the  twisting  moments  would  be  inversely  as  the  ratio  of  the  speeds 
thus: 


MACHINE  DESIGN  135 


Tw       M 


or,  T-  =  -5P  -    ^     (86) 

but  for  an  efficiency  E  the  equation  would  be: 


* 


_ 

T  =~  EN  » 


The  diameter  of  the  worm  is  arbitrary.  Change  of  thie 
diameter  has  no  effect  on  the  speed  ratio.  It  has  a  slight  effect  OL 
the  efficiency,  the  smaller  worm  giving  a  little  higher  efficiency. 
The  diameter  of  the  worm  runs  ordinarily  from  3  to  10  times  the 
circular  pitch,  an  average  value  being  4P1  or  5P1. 

A  longitudinal  cross-section  through  the  axis  of  the  worm 
cuts  out  a  rack  tooth,  and  this  tooth  section  is  usually  made  of  the 
standard  14J°  involute  form  shown  in  Fig.  46  for  a  rack. 

The  end  thrust,  of  a  mag- 
nitude practically  equal  to  the 
pressure    between    the   teeth, 
has  to  be  taken  by  the  hub  of    1    7       V—  a9*—  y       V 
the  worm  against  the  face  of     *   /  --  —  V  ---  r  --  T         / 
the  shaft  bearing.     A  serious  _  /  \  _  /  \  _  / 

loss  of  efficiency  from  friction  Fig.  46. 

is  likely  to  occur  here.     This 

is  often  reduced,  however,  by  roller  or  ball  bearings.  With  two 
worms  on  the  same  shaft,  each  driving  into  a  separate  worm  gear, 
it  is  possible  to  make  one  of  the  worms  right-hand  thread,  and 
the  other  left-hand,  in  which  case  the  thrust  is  self-contained  in 
the  shaft  itself,  and  there  is  absolutely  no  end  thrust  against  the 
face  of  the  bearing.  This  involves  a  double  outfit  throughout,  and 
is  not  always  practicable. 

There  are  few  mathematical  equations  necessary  for  the  dimen- 
sioning of  a  worm  and  worm  gear.  The  formulae  for  the  tooth 
parts  as  given  on  page  120  apply  equally  well  in  this  case. 

PRACTICAL  MODIFICATION.  The  discussion  of  the  effi- 
ciency E  of  the  worm  and  worm  gear  is  more  of  a  practical  than 


130  MACHINE  DESIGN 

of  a  theoretical  nature.  It  seems  to  be  true  from  actual  operation, 
as  well  as  theory,  that  the  steeper  the  threads  the  higher  the  effi- 
ciency. In  actual  practice  we  seldom  have  opportunity  to  change 
the  slope  of  .the  thread  to  get  increased  efficiency.  The  slope 
is  usually  settled  from  considerations  of  speed  ratio,  or  available 
space,  or  some  other  condition.  The  usual  practical  problem  is  to 
take  a  given  worm  and  worm  gear,  and  to  make  out  of  it  as  efficient 
a  device  as  possible.  With  hobbed  gears  running  in  oil  baths,  and 
with  moderate  pressures  and  speeds,  the  efficiency  will  range  between 
40  per  cent  and  70  per  cent.  The  latter  figure  is  higher  than  is 
usually  attained. 

To  avoid  cutting  and  to  secure  high  efficiency,  it  seems  es- 
sential to  make  the  worm  and  the  gear  of  different  materials. 
The  worm-thread  surfaces  being  in  contact  a  greater  number  of 
times  than  the  gear  teeth,  should  evidently  be  of  the  harder  material. 
Hence  we  usually  find  the  worm  of  steel,  and  the  gear  of  cast  iron, 
brass,  or  bronze.  To  save  the  expense  of  a  large  and  heavy  bronze 
gear,  it  is  common  to  make  a  cast-iron  center  and  bolt  a  bronze 
rim  to  it. 

The  worm,  being  the  most  liable  to  replacement  from  wear, 
it  is  desirable  so  to  arrange  its  shaft  fastening  and  general  acces- 
sibility that  it  may  be  readily  removed  without  disturbing  the 
worm  gear. 

The  circular  pitch  of  the  gear  and  the  pitch  of  the  worm 
thread  must  be  the  same,  and  the  practical  question  comes  in  as  to 
the  threads  per  inch  possible  to  be  cut  in  the  lathe  in  the  pro- 
duction of  the  worm  thread.  The  pitch  must  satisfy  this  require- 
ment; hence  the  pitch  will  usually  be  fractional,  and  the  diameter 
of  the  worm  gear,  to  give  the  necessary  number  of  teeth,  must  be 
brought  to  it.  While  it  would  perhaps  be  desirable  to  keep  an 
even  diametral  pitch  for  the  worm  gear,  yet  it  would  be  poor  de- 
sign to  specify  a  worm  thread  which  could  not  be  cut  in  a  lathe. 

The  standard  involute  of  14 J°,  and  the  standard  proportions 
of  teeth  as  given  on  page  120  are  usually  used  for  worm  threads. 
This  system  requires  the  gear  to  have  at  least  30  teeth,  for  if  fewer 
teeth  are  used  the  thread  of  the  worm  will  interfere  with  the 
flanks  of  the  gear  teeth.  This  is  a  mathematical  relation,  and 
there  are  methods  of  preventing  it  by  change  of  tooth  proportions 


MACHINE  DESIGN  137 

or  of  angle  of  worm  thread  ;  but  there  are  few  instances  in  which 
less  than  30  teeth  are  required,  and  it  is  not  deemed  worth  while 
to  go  into  a  lengthy  discussion  of  this  point. 

The  angle  of  the  worm  embraced  by  the  worm-gear  teeth 
varies  from  60°  to  90°,  and  the  general  dimensions  of  rim  are  made 
about  the  same  as  for  spur  gears.  The  arms,  or  the  web,  have  the 
same  reasons  for  their  size  and  shape.  Probably  web  gears  and 
cross-shaped  arms  are  more  common  than  oval  or  elliptical  sections. 

Worm  gears  sometimes  have  cast  teeth,  but  they  are  for  the 
roughest  service  only,  and  give  but  a  point  bearing  at  the  middle 
of  the  tooth.  An  accurately  hobbed  worm  gear  will  give  a  bearing 
clear  across  the  face  of  the  tooth,  and,  if  properly  set  up  and  cared 
for,  makes  a  good  mechanical  device  although  admittedly  of  some- 
what low  efficiency. 

Fig.  47  shows  a  detail  drawing  of  a  standard  worm  and  worm 
gear.  It  should  serve  as  a  suggestion  in  design,  and  an  illustration 
of  the  shop  dimensions  required  for  its  production. 

PROBLEMS  ON  SPUR,  BEVEL,  AND  WORM  GEARS. 

1.  Calculate  proportions    of    a    standard    Brown  &  Sharpe 
gear  tooth  of  1^  diametral  pitch,   making  a  rough  sketch  and  put- 
ting the  dimensions  on  it. 

2.  Suppose  the  above  tooth  to  be  loaded  at  the    top  with 
5,000  Ibs.     If  the  face  be  6  inches,  calculate  the  fiber  stress  at  the 
pitch  line,  due  to  bending. 

3.  A  tooth  load  of  1,200  Ibs.  is  transmitted  between  two 
spur  gears  of  12-inch  and  30-inch  diameter,  the  latter  gear  making 
100  revolutions  per  minute.     Calculate  a  suitable  pitch  and  face 
of  tooth  by  the  "  Lewis  "formula. 

4.  Assuming  a  ^-inch  web  on  the  12-inch  gear,  calculate  the 
shearing  fiber  stress  at  the  outside  of  a  hub  4  inches  in  diameter 

5.  Design  elliptical  arms    for    the    30-inch  gear,    allowing 
S  =  2,200. 

6.  Design  cross-shaped  arms  for  30-inch  gear. 

7.  Calculate  the  dimensions  shown   in  formulae  70  to  82  in- 
clusive for  a  pair  of  bevel  gears  of  20  and  60  teeth  respectively,  2 
diametral  pitch,  and  4-inch  face.     (The  use   of  logarithmic  tables 
makes  the  calculation  much  easier  than  with  the  natural  functions.) 


MACHINE  DESIGN 


8.     A  worm  wheel  has  40  teeth,  3  diametral  pitch,  and  double 
thread.     Calculate  (a)  its  lead;  (b)  its  pitch  diameter. 

FRICTION  CLUTCHES. 

ifOTATIOX— The  following  notation  is  used  throughout  the  chapter  on  Friction  Clutches :. 

a  =  Angle  between  clutch  face 
and  axis  of  shaft  (degrees) 

H  =Horse-power  (33,000  ft.-lbs. 
per  minute). 

jj,  =  Coefficient  of  friction  (per 
cent). 

N  =Number  of  revolutions  per 
minute. 

P  = Force  to  hold  clutch  in  gear 
to  produce  W  (Ibs.). 

R  =  Mean  radius  of  friction  sur- 
face (inches). 

T  =  Twisting  moment  about 
shaft  axis  (inch-lbs.). 

V  =  Force  normal  to  clutch  face 
(Ibs.). 

W  =Load  at  mean  radius  of 
friction  surface  (ibs.). 

ANALYSIS.       The 

friction  clutch  is  a  de- 
vice for  connecting  at 
will  two  separate  pieces 
of  shaft,  transmitting  an 
amount  of  power  be- 
tween them  to  the  capac- 
ity of  the  clutch.  The 
connection  is  usually  ac- 
complished while  the 
driving  shaft  is  under 
full  speed,  the  slipping 
bet  7een  the  surfaces 
which  occurs  during  the 
throwing-in  of  the 
clutch,  permitting  the 
driven  shaft  to  pick  up 
the  speed  of  the  other 
gradually,  without  ap- 
preciable shock.  The 
disconnection  is  made  in 
the  same  manner,  the 


140 


MACHINE  DESIGN" 


amount  of  slipping  which  occurs  depending  on  the  suddenness  with 
which  the  clutch  is  thrown  out. 

The  force  of  friction  is  the  sole  driving  element,  hence  the 

problem  is  to  secure  as 
large  a  force  of  friction 
as  possible.  But  friction 
cannot  be  secured  with- 
out a  heavy  normal  pres- 
sure between  surfaces 
having  a  high  coefficient 
of  friction  between  them. 
The  many  varieties  of 
friction  clutches  which 
are  on  the  market  or  de- 
signed for  some  special 
purpose,  are  all  devices 
for  accomplishing  one 
and  the  same  effect,  vis., 
the  production  of  a  heavy 
normal  force  or  pressure 
between  surfaces  at  such 
a  radius  from  the  driven 
axis,  that  the  product  of 
the  force  of  friction 
thereby  created  and  the 
radius  shall  equal  the 
desired  twisting  moment 
about  that  axis. 

Three  typical  meth- 
odsof  accomplishing  this 
are  shown  in   Figs.  48, 
49,    and   50.      None  of 
these  drawings  is  worked 
out  in  operative  detail. 
They  are    merely    illus- 
trations of  principle,  and  are  drawn  in  the  simplest  form  for  that 
purpose. 

In  Fig.  48  the  normal  pressure  is  created  in  the  simplest  pos- 


MACHINE  DESIGN 


141 


sible  way,  an  absolutely  direct  push  being  exerted  between  the 
discs,  due  to  the  thrust  P  of  the.  clutch  fork. 

In  Fig.  49  advantage  is  taken  of  the  wedge  action  of  the  in- 
clined faces,  the  result  be- 
ing that  it  takes  less  thrust 
P  to  produce  the  required 
normal  pressure  at  the  ra- 
dius K. 

In  Fig.  50  the  inclin- 
ation of  the  faces  is  carried 
so  far  that  the  angle  a  of 
Fig.  49  has  become  zero; 
and  by  the  toggle- joint  ac- 
tion of  the  link  pivoted  to 
the  clutch  collar,  the  nor- 
mal force  produced  may  be 
very  great  for  a  slight  thrust 
P.  By  careful  adjustment 
of  the  length  of  the  link  so 
that  the  jaw  takes  hold  of 
the  clutch  surface,  when 
the  link  stands  nearly  ver- 
tical, a  very  easy  operating 
device  is  secured,  and  the 
thrust  P  is  made  a  mini- 
mum. 

THEORY.  Referring 
to  Fig.  48  in  order  to  cal- 
culate the  twisting  mo- 
ment, we  must  remember 
that  the  force  of  friction 
between  two  surfaces  is 
equal  to  the  normal  pres- 
sure times  the  coefficient  of 
friction.  This,  in  the  form 
of  an  equation,  using  the  symbols  of  the  figure,  is  : 


(88) 


142  MACHINE  DESIGJT 

Hence  we   may  consider  that  we  Lave  a  force  of  magnitude 
acting  at  the  mean  radius  R  of  the  clutch  surface.     The  twisting 
moment  will  then  be  : 


T  ==  WE  ==  fjiPU.  (89) 

Referring  to  equation  54,  which  gives  twisting  moment  in  terms 
of  horse-power,  and  putting  the  two  expressions  equal  to  each  other, 
we  have  : 

63.026H 

-— 


This  expression  gives  at  once  the  horse-power  that  the  clutch  will 
transmit  with  a  given  end  thrust  P. 

In  Fig.  49  the  equilibrium  of  the  forces  is  shown  in  the  little 
sketch  at  the  left  of  the  figure.  The  clutch  faces  are  supposed 
to  be  in  gear,  and  the  extra  force  necessary  to  slide  the  two  to- 
gether is  not  considered,  as  it  is  of  small  importance.  The  static 
equations  then  are  : 

y 
P=  2-g-sin  a; 

or,  V  =  P  cosec  a.  (91) 

W  =  [jiV    =    /xP  cosec  a.  (92) 

T  =  WR  =  ^PR  cosec  a.  (93) 

'  T  =^f^  =  ^R  cosec  «; 

uNPR   cosec  a 

In  Fig.  50,  P  would  of  course  be  variable,  depending  on  the 
inclination  of  the  little  link.  The  amount  of  horse-power  which 
this  clutch  would  transmit  would  be  the  same  as  in  the  case  of  the 
device  illustrated  in  Fig.  49,  for  an  equal  normal  force  V  produced. 

The  further  theoretical  design  of  such  clutches  should  be  in 
accordance  with  the  same  principles  as  for  arms  and  webs  of 
pulleys,  gears,  etc.  The  length  of  the  hubs  must  be  liberal  in 


\ 

H=  (94) 


MACHINE  DESIGN 


order  to  prevent  tipping  on  the  shaft  as  a  result  of  uneven  wear, 
The  end  thrust  is  apt  to  be  considerable;  and  extra  side  stiffness 
must  be  provided,  as  well  as  a  rim  that  will  not  spring~untler  the 
radial  pressure. 

PRACTICAL  flODIFICATION.  It  is  desirable  to  make  the 
most  complicated  part  of  a  friction  clutch  the  driven  part,  for  then 
the  mechanism  requiring  the  closest  attention  and  adjustment  may 
be  brought  to  and  kept  at  rest  when  no  transmission  of  power  is 
desired. 

Simplicity  is  an  important  practical  requirement  in  clutches. 
The  wearing  surfaces  are  subjected  to  severe  usage;  and  it  is 
essential  that  they  be  made  not  only  strong  in  the  first  place,  but 
also  capable  of  being  readily  replaced  when  worn  out,  as  they  are 
sure  to  be  after  some  service. 

The  form  of  clutch  shown  in  Fig.  50  is  the  most  efficient 
form  of  the  three  shown,  although  its  commercial  design  is  consid- 
erably different  from  that  indicated.  Usually  the  jaws  grip  both 
sides  of  the  rim,  pinching  it  between  them.  This  relieves  the 
clutch  rim  of  the  radial  unbalanced  thrust. 

.     Adjusting  screws  must  be  provided  for  taking  up  the  wear, 
and  lock  nuts  for  maintaining  their  position. 

Theoretically,  the  rubbing,  surf  aces  should  be  of  those  materials 
whose  coefficient  of  friction  is  the  highest;  but  the  practical  ques- 
tion of  wear  comes  in,  and  hence  we  usually  find  both  surfaces  of 
metal,  cast  iron  being  most  common.  For  metal  on  metal  the 
coefficient  of  friction  p  cannot  be  safely  assumed  at  more  than  15 
per  cent,  because  the  surfaces  are  sure  to  get  oily. 

A  leather  facing  on  one  of  the  surfaces  gives  good  results  as 
to  coefficient  of  friction,  p.  having  a  value,  even  for  oily  leather,  of 
20  per  cent.  Much  slipping,  however,  is  apt  to  burn  the  leather; 
and  this  is  most  likely  to  occur  at  high  speeds. 

Wood  on  cast  iron  gives  a  little  higher  coefficient  of  friction 
for  an  oily  surface  than  metal  on  metal.  Wood  blocks  can  be  so 
set  into  the  face  of  the  jaws  as  to  be  readily  replaced  when  worn, 
and  in  such  case  make  an  excellent  facing. 

The  angle  a  of  a  cone  friction  clutch  of  the  type  shown  in 
Fig.  "49,  may  evidently  be  made  so  small  that  the  two  parts  will 
wedge  together  tightly  with  a  very  slight  pressure  P;  or  it  may 


144 


MACHINE  DESIGN 


tJD 


MACHINE  DESIGN  1-45 

be  so  large  as  to  have  little  wedging  action,  and  approach  the  con- 
dition illustrated  in  Fig.  48.  Between  these  limits  there  is  a 
practical  value  which  neither  gives  a  wedging  action  so  great  as  to 
make  the  surfaces  difficult  to  pull  apart,  nor,  on  the  other  hand, 
requires  an  objectionable  end  thrust  along  the  shaft  in  order  to 
make  the  clutch  drive  properly. 

For  a  =  about  15°,  the  surfaces  will  free  themselves  when  P  is  relieved. 
"    a  =      "       12°,    "         "         require  slight  pull  to  be  freed. 
"    a  =      "      10°,    "         "         cannot  be  freed  by  direct  pull  of  the 
hand,  but  require  some  leverage  to  produce  the  necessary  force  P. 

PROBLEMS  ON  FRICTION   CLUTCHES. 

1.  With  what  force  must   we  hold   a  friction  clutch  in  to 
transmit    30  horse-power  at  200  revolutions  per  minute,  assuming 
working  radius  of  clutch  to  be  12  inches;  coefficient  of  friction  15 
per  cent;  angle  a  =  10°  ? 

2.  How  much  horse-power  could  be  transmitted,  other  con- 
ditions remaining  the  same,  if  the  working  radius  were  increased 
to  18  inches  ? 

3.  What  force  would  be  necessary  in  problem  1,  if  the  angle 
a  were  15°,  other  conditions  remaining  the  same  ? 

COUPLINGS. 

(STOTATION.—  The  following  notation  is  used  throughout  the  chapter  on  Couplings: 

D  =  Diameter  of  shaft  (inches).  Sc  =  Safe  crushing  fiber  stress  (Ibs.  per 
d    =  Diameter  of  bolt  body  (inches).  sq.  in.), 

n   =Number  of  bolts.  T  =Twisting  moment  (inch-lbs.). 
R  =Radius  of  bolt  circle  (inches).  £=Th'ickness  of  flange  (inches). 

S  =Safe  shearing  fiber  stress  (Ibs.  per  W=Load  on  bolts  (Ibs.). 
sq.  in.). 

ANALYSIS.  Rigid  couplings  are  intended  to  make  the 
shafts  which  they  connect  act  as  a  solid,  continuous  shaft.  In 
order  that  the  shaft  may  be  worked  up  to  its  full  strength  capac- 
ity, the  coupling  must  be  as  strong  in  all  respects  as  the  shaft, 
or,  in  other  words,  it  must  transmit  the  same  torsional  moment. 
In  the  analysis  of  the  forces  which  come  upon  these  couplings,  it 
is  not  considered  that  they  are  to  take  any  side  load,  but  thrt  they 
are  to  act  purely  as  torsional  elements.  It  is  doubtless  true  that  in 
many  cases  they  do  have  to  provide  some  side  strength  and  stiff- 
ness, but  this  is  not  their  natural  function,  nor  the  one  upon  which 
their  design  is  based. 


146  MACHINE  DESIGN 

Referring  to  Fig.  51,  which  is  the  type  most  convenient  for 
analysis,  we  have  an  example  of  the  simplest  form  of  flange  coup- 
ling. It  consists  merely  of  hubs  keyed  to  the  two  portions,  with 
flanges  driving  through  shear  on  a  series  of  bolts  arranged  con- 
centrically about  the  shaft.  The  hubs,  keys,  and  flanges  are  sub- 
ject to  the  same  conditions  of  design  as  the  hubs,  keys,  and  web  of 
a  gear  or  pulley,  the  key  tending  to  shear  and  be  crushed  in  the 
hub  and  shaft,  and  the  hub  tending  to  be  torn  or  sheared  from  the 
flange.  The  driving  bolts,  which  must  be  carefully  fitted  in 
reamed  holes,  are  subject  to  a  purely  shearing  stress  over  their  full 
area  at  the  joint,  and  at  the  same  time  tend  to  crush  the  metal  in 
the  flange,  against  which  they  bear,  over  their  projected  area. 
This  latter  stress  is  seldom  of  importance,  the  thickness  of  the 
flange,  for  practical  reasons,  being  sufficient  to  make  the  crushing 
stress  very  low. 

THEORY.  The  theory  of  hubs,  keys,  and  flanges,  being  like 
that  already  given  for  pulleys  and  gears,  need  not  be  repeated  for 
couplings.  The  shearing  stress  on  the  bolts  is  the  only  new  point 
to  be  studied. 

In  Fig.  51,  for  a  twisting  moment  on  the  shaft  of  T,  the  load 

T 

at    the  bolt    circle  is  W  =  -=&-•       If  the  number  of  bolts  be  n, 

equating  the  external  force  to  the  internal  strength,  we  have: 

T. 


Although  the  crushing  will  seldom  be  of  importance,  yet  for 
the  sake  of  completeness  its  equation  is  given,  thus: 


W  =        ==  Scdtn.  (96) 

SD3 

The  internal    moment    of    resistance    of   the    shaft    is  -=^-  \ 

O.-L 

hence  the  equation  representing  full  equality  of  strength  between 
the  shaft  and  the  coupling,  depending  upon  the  shearing  strength 
of  the  bolts,  is  : 

SD3 


MACHINE  DESIGN 


147 


The  theory  of  the  other  types  of  couplings  is  obscure,  except 
as  regards  the  proportions  of  the  key,  which  are  the  same  in  all 
cases.  The  shell  of  the  clamp  coupling,  Fig.  52,  shoukl  be  thick 
enough  to  give  equal  torsional  strength  with  the  shaft;  but  the 
exact  function  which  the  bolts  perform  is  difficult  to  determine. 
In  general  the  bolts  clamp  the  coupling  tightly  on  the  shaft  and 
provide  rigidity,  but  the  key  does  the  principal  amount  of  the 
driving.  The  bolt  sizes,  in  these  couplings,  are  based  on  judgment 
and  relation  to  surrounding  parts,  rather  than  on  theory. 

PRACTICAL  HODIFICATION.  All  couplings  mus*  be  made 
with  care  and  nicely  fitted,  for  their  tendency,  otherwise,  is  to 


Fig.  52. 

spring  the  shafts  out  of  line.  In  the  case  of  the  flange  coupling, 
the  two  halves  may  be  keyed  in  place  on  the  shafts,  the  latter  then 
swung  on  centers  in  the  lathe,  and  the  joint  faced  off.  Thus  the 
joint  will  be  true  to  the  axis  of  the  shaft;  and,  when  it  is  clamped 
in  position  by  the  bolts,  no  springing  out  of  line  can  take  place. 

A  flange  F  (see  Fig.  51*)  is  sometimes  made  on  this  form  of 
coupling,  in  order  to  guard  the  bolts.  It  may  be  used,  also,  to  take 
a  light  belt  for  driving  machinery;  but  a  side  load  is  thereby  thrown 
on  the  shaft  at  the  joint,  which  is  at  the  very  point  whore  it  is  desir- 
able to  avoid  it. 

The  simplest  form  of  rigid  coupling  consists  of  a  plain  sleeve 
slipped  over  from  one  shaft  to  the  other,  when  the  second  is  butted 
up  against  the  first.  This  is  known  as  a  muff  coupling.  When 
once  in  place,  this  is  a  very  excellent  coupling,  as  it  is  perfectly 
smooth  on  the  outside,  and  consists  of  the  fewest  possible  parts, 
merely  a  sleeve  and  a  key.  It  is,  however,  expensive  to  fit, 


148 


MACHINE  DESIGN 


difficult  to  remove,  and  requires  an  extra  space  of  half  its  length 
on  the  shaft  over  which  to  be  slipped  back. 

The  clamp  coupling  is  a  good  coupling  for  moderate -sized 
shafts,  where  the  flange  type  of  Fag.  51  would  be  unnecessarily 
expensive.  The  clamp  coupling,  Fig.  52,  is  simply  a  muff  coupling 
split  in  halves,  and  recessed  for  bolts.  It  is  cheap  and  is  easily 
applied  and  removed,  even  with  a  crowded  shaft.  If  bored  with  a 
piece  of  paper  in  the  joint,  when  it  is  clamped  in  position  it  will 
pinch  the  shaft  tightly  and  make  a  rigid  connection.  It  is  desir- 
able to  have  the  bolt-heads  protected  as  much  as  possible,  and  this 
may  be  accomplished  by  making  the  outside  diameter  large  enough 
so  that  the  bolts  will  not  project.  Often  an  additional  shell  is 
provided  to  encase  the  coupling  completely  after  it  is  located. 


n 


Fig.  53. 

There  are  many  other  special  forms  of  couplings,  some  of 
them  adjustable.  Most  of  them  depend  upon  a  wedging  action 
exerted  by  taper  cones,  screws,  or  keys.  Trade  catalogues  are  to 
be  sought  for  their  description. 

The  claw  coupling,  Fig.  53,  is  nothing  but  a  heavy  flange 
coupling  with  interlocking  claws  or  jaws  on  the  faces  of  the  flanges, 
to  take  the  place  of  the  driving  bolts.  This  coupling  can  be  thrown 
in  or  out  as  desired,  although  it  usually  performs  the  service  of  a 
rigid  coupling,  as  it  is  not  suited  to  clutching-in  during  rapid  mo- 
tion, like  a  friction  clutch. 

Flexible  couplings,  which  allow  slight  lack  of  alignment,  are 
made  by  introducing  between  the  flanges  of  a  coupling  a  flexible 
disc,  the  one  flange  being  fastened  to  the  inner  circle  of  the  disc, 
the  other  to  the  outer  circle.  This  is  also  accomplished  by  pro- 
viding the  faces  of  the  flange  coupling  with  pins  that  driye  by 


MACHINE  DESIGN 


149 


pressure  together  or  through  leather  straps  wrapped  round  the 
pins.  These  devices  are  mostly  of  a  special  and  often  uncertain 
nature,  lacking  the  positiveness  which'  is  one  essential  feature 
of  a  good  coupling. 

PROBLEMS  ON  COUPLINGS. 

L  A  flange  coupling  of  the  type  of  Fig.  51  is  usea  on  a  shaft 
2  inches  is  diameter.  The  hub  is  3  inches  long,  and  carries  a 
standard  key?  of  proportions  indicated  below  in  the  table  of  "Pro- 
portions for  Gib  Keys"  (page  166).  The  bolt  circle  is  7  inches 
in  diameter,  and  it  is  desired  to  use  ft-inch  bolts.  How  many 

O  e/ 

bolts  are  needed  to  transmit  60,000  inch-lbs.,  for  a  fiber  stress  in 
the  bolt  of  6,000  ? 

2.  Using  6  bolts,  what  diameter  of  bolt  would  be  required  ? 

3.  If  four  £-inch  bolts  were  used  on  a  circle  of  8  inches  di- 
ameter, what  diameter  of  shaft  would  be  used   in  the  coupling  to 
give  equal  strength  with  the  bolts  ? 

BOLTS,  STUDS,  NUTS,  AND  SCREWS. 

NOTATION— The  following  notation  is  used   throughout  the  chapter  on  Bolts,  Studs, 

Nuts,  and  Screws: 


d   =  Diameter  of  bolt  (inches). 

dl  =  Diameter  at  root  of  thread  (in- 
.ches). 

H  '=  Height  of  nut  (inches). 

I    =  Initial  axial  tension  (Ibs.). 

k   =  Allowable  bearing  pressure  on  sur- 
face of  thread  (Ibs,  per  sq.  in.). 

L  =  Lead,  or  distance  nut  advances 
along  axis  in  one  revolution 
(inches). 


I      —  "Length  of  wrench  handle  (inches) 

H 

n     —  Number  of  threads  in  nut  =  — . 

P 

P    =  Axial  load  (Ibs.). 

p  —-Pitch  of  thread,  or  distance  be- 
tween similar  points  on  adjacent 
threads,  measured  parallel  to 
axis  Cinches). 

S    =  Fiber  stress  (Ibs.  per  sq.  in.). 

W  =  Load  on  bolt  (Ibs.). 


Fig.   54. 

ANALYSIS.  A  bolt  is  simply  a  cylindrical  bar  of  metal 
upset  at  one  end  to  form  a  head,  and  having  a  thread  at  the  other 
end,  Fig.  54.  A  stud  is  a  bolt  in  which  the  head  is  replaced  by 
a  thread;  or  it  is  a  cylindrical  bar  threaded  at  both  ends,  usually 


150 


MACHINE   DESIGN 


having  a  small  plain  portion  in  the  middle,  Fig.  55.  The  object 
of  bolts  and  studs  is  to  clamp  machine  parts  together,  and  yet 
permit  these  same  parts  to  be  readily  disconnected.  The  bolt 
passes  through  the  pieces  to  be  connected,  and,  when  tightened, 
causes  surface  compression  between  the  parts,  while  the  reactions 
on  the  head  and  nut  produce  tension  in  the  bolt.  Studs  and  tap 
bolts  pass  through  one  of  the  connected  parts  and  are  screwed 
into  the  other,  the  stud  remaining  in  position  when  the  parts  are 
disconnected. 

As  all  materials  are  elastic  within  certain  limits,  the  action  of 


Fig.  55. 


Fig.  55a. 


a  bolt  in  clamping  two  machine  parts  together,  more  especially  if 
there  is  an  elastic  packing  between  them,  may  be  represented 
diagrammatically  by  Fig.  56,  in  which  a  spring  has  been  introduced 
to  take  the  compression  due  to  screwing  up  the  nut.  Evidently 
the  tension  in  the  bolt  is  equal  to  the  force  necessary  to  compress 
the  spring.  Now,  suppose  that  two  weights,  each  equal  to  ^  W, 
are  placed  symmetrically  on  either  side  of  the  bolt,  then  the  tension 
in  the  bolt  will  be  increased  by  the  added  weights  if  the  bolt  is 
perfectly  rigid.  The  bolt,  however,  stretches;  hence  some  of  the 
compression  on  the  spring  is  relieved  and  the  total  tension  in  the 


MACHINE  DESIGN 


151 


bolt  is  less  than  W  +  I,  by  an  amount  depending  on  the  relative 
elasticity  of  the  bolt  and  spring. 

Suppose  that  the  stud  in  Fig.  55  is  one  of  the  studs_connect- 
ing  the  cover  to  the  cylinder  of  a  steam  engine,  and  that  the  studs 
have  a  small  initial  tension;  then  the  pressure  of  the  steam  loads 
each  stud,  and,  if  the  studs  stretch,  enough  to  relieve  the  initial 
pressure  between  the  two  surfaces,  then  their  stress  is  due  to  the 
steam  pressure  only;  or,  from  Fig.  56,  when  I  =  W  ;  the  initial 
pressure  due  to  the  elasticity  of  the  joint  is  entirely  relieved  by  the 
assumed  stretch  of  the  studs.  Except  to  prevent  leakage,  it  is 
seldom  necessary  to  consider  the  initial  tension,  for  the  stretch 
'of  the  bolt  may  be  counted  on  to  relieve 
this  force,  and  the  working  tension  on  the 
bolt  is  simply  the  load  applied. 

For  shocks  or  blows,  as  in  the  case 
of  the  bolts  found  on  the  marine  type  of 
connecting-rod  end,  the  stretch  of  the 
bolts  acts  like  a  spring  to  reduce  the  re- 
sulting tensions.  So  important  is  this 
feature  that  the  body  of  the  bolt  is  fre- 
quently turned  down  to  the  diameter  of 
the  bottom  of  the  thread,  thus  uniformly 
distributing  the  stretch  through  the  full 
lengfehof  the  bolt,  instead  of  localizing  it 
at  the  threaded  parts. 

In  tightening  up  a  bolt,  the  friction 
at  the  surface  of  the  thread  produces  a  twisting  moment,  which 
increases  the  stress  in  the  bolts,  just  as  in  the  case  of  shafting 
under  combined  tension  and  torsion;  but  the  increase  is  small  in 
amount,  and  may  readily  be  taken  care  of  by  parrnitting  low  values 
only  for  the  fiber  stress. 

In  a  flange  coupling,  bolts  are  acted  upon  by  forces  perpen- 
dicular to  the  axis,  and  hence  are  under  pure  shearing  stress.  If 
the  torque  on  the  shaft  becomes  too  great,  failure  will  occur  by 
the  bolts  shearing  off  at  the  joint  of  the  coupling. 

A  bolt  under  tension  communicates  its  load  to  the  nut  through 
the  locking  of  the  threads  together.  If  the  nut  is  thin,  and  the 
number  of  threads  to  take  the  load  few,  the  threads  may  break  or 


Fig.  56. 


152  MACHINE  DESIGN 

shear  off  at  the  root.  With  a  Y  thread  there  is  produced  a  com- 
ponent force,  perpendicular  to  the  axis  of  the  bolt,  which  tends  to 
split  the  nut. 

In  screws  for  continuous  transmission  of  motion  and  power, 
the  thread  may  be  compared  to  a  rough  inclined  plane,  on  which  a 
small  block,  the  nut,  is  being  pushed  upward  by  a  force  parallel  to 
the  base  of  the  plane.  The  angle  at  the  bottom  of  the  plane  is  the 
angle  of  the  helix,  or  an  angle  whose  tangent  is  the  lead  divided 
by  the  circumference  of  the  screw.  The  horizontal  force  corre- 
sponds to  the  tangential  force  on  the  screw.  The  friction*  at  the 
surface  of  the  thread  produces  a  twisting  moment  about  the  axis  of 
the  screw,  which,  combined  with  the  axial  load,  subjects  the  screw 
to  combined  tension  and  torsion.  Screws  with  square  threads  are 
generally  used  for  this  service,  the  sides  of  the  thread  exerting  no 
bursting  pressure  on  the  nut.  The  proportions  of  screw  thread 
for.  transmission  of  power  depend  more  on  the  bearing  pressure 
than  on  strength.  If  the  bearing  surface  be  too  small  and  lubrica- 
tion poor,  the  screw  will  cut  and  wear  rapidly. 

THEORY.  A  direct  tensile  stress  is  induced  in  a  bolt  when 
it  carries  a  load  exerted  along  its  axis.  This  load  must  be  taken 
by  the  section  of  the  bolt  at  the  bottom  of  the  thread.  If  the  area 

at  the  root  of  the  thread  is  —  —  !—  ,  and  if  S  is  the  allowable  stress 

per    square    inch,    then    the    internal    resistance    of   the    bolt    is 

—  .    Equating  the  external  load  to  the  internal  strength  we  have: 


For  bolts  which  are  used  to  clamp  two  machine  parts  together 
so  that  they  will  not  separate  under  the  action  of  an  applied  load, 
the  initial  tension  of  the  bolt  must  be  at  least  equal  to  the  applied 
load.  If  the  applied  load  is  W,  then  the  parts  are  just  about  to 
separate  when  I  =  "W.  Therefore  the  above  relation  for  strength 
is  applicable.  As  the  initial  tension  to  prevent  separation  should 
be  a  little  greater  than  W,  a  value  of  S  should  be  chosen  so  that 
there  will  be  a  margin  of  safety.  For  ordinary  wrought  iron  and 
steel,  S  may  be  taken  at  6,000  to  8,000. 


MACHINE  DESIGN 


153 


If,  however,  the  joints  must  be  such  that  there  is  no  leakage 
between  the  surfaces,  as  in  the  case  of  a  steam  cylinder  head,  and 
supposing  that  elastic  packings  are  placed  in  the  joints,  then  a 
much  larger  margin  should  be  made,  for  the  maximum  load  which 
may  come  on  the  bolt  is  I  -4-  W,  where  W  is  the  proportional 
share  of  the  internal  pressure  carried  by  the  bolt.  In  such  cases 
S  —  3,000  to  5,000,  using  the  lower  value  for  bolts  of  less  than 
|-inch  diameter. 

The  table  given  on  page  154  will  be  found  very  useful  in  pro- 
portioning bolts  with  U.  S.  standard  thread  for  any  desired  fiber 
stress. 

To  find  the  initial  tension  due  to  screwing  up  the  nut,  we 


Fig.  56a. 

may  assume  the  length  of  the  handle  of  an  ordinary  wrench,  meas- 
ured from  the  center  of  tha  bolt,  as  about  16  times  the  diameter 
of  the  bolt.  For  one  turn  of  the  wrench  a  force  F  at  the  handle 
would  pass  over  a  distance  2?r/,  and  the  wrork  done  is  equal  to  the 
product  of  the  force  and  space,  or  F  X  2irL  At  the  same  time 
the  axial  load  P  would  be  moved  a  distance  p  along  the  axis. 
Assuming  that  there  is  no  friction,  the  equation  for  the  equality 
of  the  work  at  the  handle  and  at  the  screw  is: 


F27TZ  = 


(99) 


Friction,  however,  is  always  present;  hence  the  ratio  of  the  useful 
work  (Pp)  to  the  work  applied  (¥2irl)  is  not  unity  as  above  re- 


02  H  g 

|l<2 

S  3  h 

*s° 

ssi 
«  33 


•ui 


•bis  aad 
il  OOO'i 
IV 


•ut   -bs  J9d 
•sqi  000'9 

IV 


SSSS3 


•ut   -bs  J9d 

•sqi   OOO'S 

IV 


-bs  aad 

i  ooo> 


t- 


- 
SS 


M-i 

H 

O       O 

g«s 
s^z 

Is 


HgP 
MKr° 
«EH 


•ut  -bs  -aed 
•SQT  OOO'Ol 

av 


oi  M<  oo  cc  o 


•ut   -bs  aad 

•sqi  OOO'Z, 

IV 


•ut   -bs 


i  000'9 
IV 


•ut 


-bs  aad 
i  OOO'Q 
IV 


•ut  *bs  aad 

•sqi  000'^ 

IV 


•p^gaqj, 
jo  uioi-jog: 


^ioe 


TTWI 


jo  raoc^oe 


•qout 


MACHINE  DESIGN 


156 


lations  assume.  From  numerous  experiments  on  the  frict'on  of 
screws  and  nuts,  it  has  been  found  that  the  efficiency  may  be  as 
low  as  10  per  cent.  Introducing  the  efficiency  in  above  equation, 
it  may  be  written: 

P  1 


Assuming    that    50    pounds   is    exerted    by    a    workman    in 


Fig.  58. 

tightening  up  the  nut  on  a  1-inch  bolt,  the  equation  above  shows 
that  P=  4,021  pounds;  or  the  initial  tension  is  somewhat  less 
than  the  tabular  safe  load  shown  for  a  1-inch  bolt,  with  S  assumed 
at  10,000  pounds  per  sq.  inch. 

For  shearing  stresses  the  bolt  should  be  fitted  so  that  the  body 
of  the  bolt,  not  the  threads,  resists  the  force  tending  to  shear  off 

o 

the  bolt  perpendicular  to  its  axis.  The  internal  strength  of  the 
bolt  to  resist  shear  is  the  allowable  stress  S  times  the  area  of  the 

O  72 

bolt  in  shear,  or —    — .     If  W  represents  the  external  force  tending 

to  shear  the  bolt  the  equality  of  the  external  force  to  the  internal 
strength  is  : 

2 

(ioi) 


156  MACHINE  DESIGN 

Reference  to  the  table  on  page  154  for  the  shearing  strength  of 
bolts,  may  be  made  to  save  the  labor  of  calculations. 

Let  Fig.  58  represent  a  square  thread  screw  for  the  transmis- 
sion of  motion.     The  surface  on  which  the  axial  pressure  bears,  if 

n  is  the  number  of  threads  in  a  nut,  is  —  (d?  —  d^)  n.     Suppose 

that  a  pressure  of  k  pounds  per  square  inch  is  allowed  on  the 
surface  of  the  thread.  Then  the  greatest  permissible  axial  load  P 
must  not  exceed  the  allowable  pressure;  or,  equating, 


P  =  i-  JJ.  (ffi  -  d{)  n  (,02) 


The  value  of  k  varies  with  the  service  required.  If  the  motion  be 
slow  and  the  lubrication  very  good,  k  may  l>e  as  high  as  900.  For 
rapid  motion  and  doubtful  lubrication,  k  may  not  be  over  200. 
Between  these  extremes  the  designer  must  use  his  judgment, 
remembering  that  the  higher  the  speed  the  lower  is  the  allowable 
bearing  pressure. 

PRACTICAL  MODIFICATION.  It  will  be  noted  in  the 
formulae  for  bolt  strengths  that  different  values  for  S  are  assumed. 
This  is  necessary  on  account  of  the  uncertain  initial  stresses  which 
are  produced  is  setting  up  the  nuts.  For  cases  of  mere  fastening, 
the  safe  tension  is  high,  as  just  before  the  joint  opens  the  tension 
is  about  equal  to  the  load  and  yet  the  fastening  is  secure.  On 
the  other  hand,  bolts  or  studs  fastening  joints  subjected  to  internal 
fluid  pressure  must  be  stressed  initially  to  a  greater  amount  than 
the  working  pressure  which  is  to  come  on  the  bolt.  As  this  initial 
stress  is  a  matter  of  judgment  on  the  part  of  the  workman,  the 
designer,  in  order  to  be  on  the  safe  side,  should  specify  not  less 
than  I -inch  or  |-inch  bolts  for  ordinary  work,  so  that  the  bolts 
may  not  be  broken  off  by  a  careless  workman  accidentally  putting 
a  greater  force  than  necessary  on  the  wrench  handle.  In  making 
a  steam-tight  joint,  the  spacing  of  the  bolts  will  generally  deter- 
mine their  number;  hence  we  often  find  an  excess  of  bolt  strength 
in  joints  of  this  character. 

..  Through  bolts  are  preferred  to  studs,  and  studs  to  tap  bolts 
or  cap  screws.  If  possible,  the  design  should  be  such  that  througn 
bolts  may  be  used.  They  are  cheapest,  are  always  in  standard 


MACHINE  DESIGN  157 

stock,  and  well  resist  rough  usage  in  connecting  and  disconnecting. 
The  threads  in  cast  iron  are  weak  and  have  a  tendency  to  crumble; 
and  if  a  through  bolt  cannot  be  used  in  such  a  case,  a^stud,_ which 
can  be  placed  in  position  once  for  all,  should  be  employed— not  a 
tap  bolt,  which  injures  the  thread  in  the  casting  every  time  it  is 
removed. 

The  plain  portion  of  a  stud  should  be  screwed  up  tight 
against  the  shoulder,  and  the  tapped  hole  should  be  deep  enough 
to  prevent .  bottoming.  To  avoid  breaking  off  the  stud  at  the 
shoulder,  a  neck,  or  groove,  may  be  made  at  the  lower  end  of  the 
thread  entering  the  nut. 

To  withstand  shearing  forces  the  bolts  must  be  fitted  so  that 
no  lost  motion  may  occur,  otherwise  pure  shearing  will  not  be 
secured. 

Nuts  are  generally  made  hexagonal,  but  for  rough  work  are 
often  made  square.  The  hexagonal  nut  allows  the  wrench  to  turn 
through  a  smaller  angle  in  tightening  up,  and  is  preferred  to  the 
square  nut.  Experiments  and  calculations  show  that  the  height 
of  the  nut  with  standard  threads  may  be  about  -J  the  diameter  of 
the  bolt  and  still  have  the  shearing  strength  of  the  thread  equal  to 
the  tensile  strength  of  the  bolt  at  the  root  of  the  thread.  Practi- 

o 

cally,  however,  it  is  difficult  to  apply  such  a  thin  wrench  as  this 
proportion  would  call  for  on  ordinary  bolts.  More  commonly  the 
height  of  the  nut  is  made  equal  to  the  diameter  of  the  bolt  so  that 
the  length  of  thread  will  guide  the  nut  on  the  bolt,  give  a  low 
bearing  pressure  on  the  threads,  and  enable  a  suitable  wrench  to 
be  easily  applied.  The  standard  proportions  for  bolts  and  nuts 
may  be  found  in  any  handbook.  Not  all  manufacturers  conform 
to  the  United  States  standard;  nor  do  manufacturers  in  all  cases 
conform  to  one  another  in  practice. 

If  the  bolt  is  subject  to  vibration,  the  nuts  have  a  tendency  to 
loosen.  A  common  method  of  preventing  this  is  to  use  double 
nuts,  or  lock  nuts,  as  they  are  called  (see  Fig.  55  A).  The  under 
nut  is  screwed  tightly  against  the  surface,  and  held  by  a  wrench 
while  the  second  nut  is  screwed  down  tightly  against  the  first. 
The  effect  is  to  cause  the  threads  of  the  upper  nut  to  bear  against 
the  under  sides  of  the  threads  of  the  bolt.  The  load  on  the  bolt  is 
sustained  therefore  by  the  upper  nut,  which  should  be  the  thicker 


158 


MACHINE  DESIGN 


of  the  two  ;  but  for  convenience  in  applying  wrenches  the  position 
of  the  nuts  is  often  reversed. 

The  form  of  thread  adapted  to  transmitting  power  is  the 
square  thread,  which,  although  giving  less  bursting  pressure 
on  the  nut,  is  not  as  strong  as  the  Y  thread  for  a  given  length, 
since  the  total  section  of  thread  at  the  bottom  is  only  ^  as  great. 
If  the  pressure  is  to  be  transmitted  in  but  one  direction,  the  two 


Fig.  59. 

types  may  be  combined  advantageously  to  form  the  buttress  thread 
of  the  proportions  shown  in  Fig.  59.  Often,  as  in  the  carriage  of 
a  lathe,  to  allow  the  split  nut  to  be  opened  and  closed'over  the  lead 
screw,  the  sides  of  the  thread  are  placed  at  a  small  angle,  say  15°, 
to  each  other,  as  illustrated  in  Fig.  60. 

The  practical  commercial  forms  in  which  we   find  screwed 
fastenings  are  included  in  five  classes,  as  follows: 


-CH 


Fig.  60. 

1.  Through  bolts  (Fig.  61),  usually  rough  stock,  with  square 
upset  heads,  and  square  or  hexagonal  nuts. 

2.  Tap  bolts  (Fig.  62),  also  called  cap  screws.     These  usu- 
ally have  hexagonal  heads,  and  are  found  both  in  the  rough  form, 
and  finished  from  the  rolled  hexagonal  bar  in  the  screw  machine. 

3.  Studs  (Fig.  68),  rough  or  finished  stock,  threaded  in  the 
screw  machine. 

4.  Set  screws   (Fig.  64),  usually  with  square    heads,  and 
case-hardened  points.     Many  varieties  of  set  screws  are  made,  the 


MACHINE  DESIGN 


159 


principal  distinguishing  feature  of  each  being  in  the  shape  of  the 
point.  Thus,  in  addition  to  the  plain  beveled  point,  we  find 
the  "  cupped,"  rounded,  conical,  and  "  teat "  points. 


Fig.  61. 


Fig.  62. 


Fig.  63. 

5.  Machine  screws  (Fig.  64a),  usually  round,  "  button," 
or  countersunk  head.  Common  proportions  are  indicated  relative 
to  diameter  of  body  of  screw. 


Fig.  64. 

PROBLEflS  ON  BOLTS,  STUDS,  NUTS,  AND  SCREWS. 

1.     Calculate    the   diameter  of   a  bolt   to  sustain  a  load  of 
5,000  Its. 


160 


MACHINE  DESIGN 


2.  The  shearing  force  to  be  resisted  by  each  of  the  bolts  of  a 
flange  coupling  is  1,200  Ibs.     What  commercial   size  of  bolt  is 
required  ?  4 

3.  With  a  wrench  16  times  the  diameter  of  the  bolt,  and  an 
efficiency  of  10  per  cent,  what  axial  load  can  a  man  exert  on  a 
standard  £-inch  bolt,  if  he  pulls  40  Ibs.  at  the  end  of  the  wrench 
handle  ? 

4.  A    single,  square -threaded  screw  of  diameter   2  inches, 
lead  ^  inch,  depth  of  thread  J  inch,  length  of  nut  3  inches,  is  to 
be  allowed  a  bearing  pressure  of  300  Ibs.  per  square  inch.     What 
axial  load  can  be  carried  ? 

5.  Calculate  the  shearing  stress  at  the  root  of  the  thread  in 
problem  4. 


Fig.  64a. 

KEYS,  PINS,  AND  COTTERS. 

NOTATION— The  following  notation  is  used  throughout  the  chapter  on  Keys,  Pins,  and 

Cotters: 


D  =  Average  diameter  of  rod  (inches). 
Di  =  Outside  diameter  of    socket  (in- 

c'aes). 

d    =  Diameter  of  shaft  (inches). 
Li  =  Length  of  key  (inches). 
P  =  Driving  force  (Ibs.). 
PI  =  Axial  load  on  rod  (Ibs.). 
R  =  Radius  at  which  P  acts  (inches). 
Sc  =  Safe  crushing  fiber  stress  (Ibs. 

per  sq.  in.). 


S^  =  Safe  shearing  fiber  stress  (Ibs. 
per  sq.  in.). 

St  =Safe  tensile  fiber  stress  (Ibs.  per 
sq.  in.). 

T  =  Thickness  of  key  (inches). 

W  =  Width  of  key  (inches). 

w  =  Average  width  of  cotter  (inches). 

v)\  =  End  of  slot  to  end  of  rod  (inches). 

W2  =  End  of  slot  to  end  of  socket  (in- 
ches). 


KEYS  AND  PINS. 
ANALYSIS.     Keys    and    pins   are    used  to  prevent   relative 


MACHINE  DESIGN  161 


rotary  motion  between  machine  parts  intended  to  act  together  as 
one  piece.  If  we  drill  completely  through  a  hub  and  across  the 
shaft,  and  insert  a  tightly  fitted  pin,  any  rotary  motion  G£  the  one 
will  be  transmitted  to  the  other,  provided  the  pin  does  not  fail  by 
shearing  off  at  the  joint  between  the  shaft  and  the  hub.  The 
shearing  area  is  the  sum  of  the  cross -sections  of  the  pin  at  the 
joint. 

We  may  drill  a  hole  in  the  joint,  the  axis  of  the  hole  being 
parallel  to  the  axis  of  the  shaft,  and  drive  in  a  pin,  in  which  case 
wre  introduce  a  shearing  area  as  before,  but  the  area  is  now  equal 
to  the  diameter  of  the  pin  multiplied  by  its  length,  and  the  pin  is 
stressed  sidewise,  instead  of  across.  It  is  evident  in  the  sidewise 
case  that  we  may  increase  the  shearing  area  to  anything  we  please, 
without  changing  the  diameter  of  the  pin,  merely  by  increasing 
the  length  of  the  pin. 

As  there  are  some  manufacturing  reasons  why  a  round  pin 
placed  lengthwise  in  the  joint  is  not  always  applicable,  we  may 
make  the  pin  a  rectangular  one,  in  which  case  it  is  called  a  key. 

When  pins  are  driven  across  the  shaft  as  in  the  first  instance, 
they  are  usually  made  taper.  This  is  because  it  is  easier  to  ream 
a  taper'hole  to  size  than  a  straight  hole,  and  a  taper  pin  will  drive 
more  easily  than  a  straight  pin,  it  not  being  necessary  to  match  the 
hole  in  hub  and  shaft  so  exactly  in  order  that  the  pin  may  enter. 
The  taper  pin  will  draw  the  holes  into  line  as  it  is  driven,  and  can 
be  backed  out  readily  in  removal. 

Keys  of  the  rectangular  form  are  either  straight  or  tapered, 
but  for  different  reasons  from  those  just  stated  for  pins.  Straight 
keys  have  working  bearing  only  at  the  sides,  driving  purely  by 
shear,  crushing  being  exerted  by  the  side  of  the  key  in  both  shaft 
and  hub,  over  the  area  against  the  key.  The  key  itself  does  not 
prevent  end  motion  along  the  shaft;  and  if  end  motion  is  not 
desired,  auxiliary  means  of  some  sort  must  be  resorted  to,  as,  for 
example,  set  screws  through  the  hub  jamming  hard  against  the  top 
of  the  key. 

If  end  motion  along  the  shaft  is  desired,  the  key  is  called  a 
spline,  and,  while  not  jammed  against  the  shaft,  is  yet  prevented 
from  changing  its  relation  to  the  hub  by  some  means  such  as 
illustrated  in  Fig.  65. 


162 


MACHINE  DESIGN 


Taper  keys  not  only  drive  through  sidewise  shearing  strength, 
but  prevent  endwise  motion  by  the  wedging  action  exerted  between 
the  shaft  and  hub.  These  keys  drive  more  like  a  strut  from  corner  to 
corner;  but  this  action  is  incidental  rather  than  intentional,  and  the 
proportions  of  a  taper  key  should  be  such  that  it  will  give  its  full 
resisting  area  in  shearing  and  crushing,  the  same  as  a  straight  key. 


Fig.  65. 

THEORY.     Suppose  that  the  pin  illustrated  in  Fig.  6' 
through  hub  and  shaft,  and  the  driving  force  P  acts  at  the  radius 
K;  then  the  force  which  is  exerted  at  the  surface  of  the  shaft  to 

2  PR 
shear  off  the  pin  at  the  points  A  and  B  is   -      ,     .     If  Dj  is  the 


average  diameter  of  the  pin,  its  shearing  strength  is 


27rD12S8 


Equating  the  external  force  to  the  internal  strength,  we  have  : 

2PR       27TD,2  Ss 
4~ 

4PR~ 


or, 


TrtZS. 


(103) 


In  Fig.  67  a  rectangular  key  is  sunk  half  way  in  hub  and 
shaft  according  to  usual  practice.     Here  the  force  at  the  surface 


MACHINE  DESIGN 


163 


of  the  shaft,  .calculated  the  same  as  before,  not  only  tends  to  shear 
off  the  key  along  the  line  AB,  but  tends  to  crush  both  the  por- 
tion in  the  shaft  and  in  the  hub.  The  shearing  strength_along  the 


Fig.  66. 


Fig.  67. 


line  AB  is  LWSS.     Equating  external  force  to  internal  strength, 
we  have: 

2PK 


or. 


2PK 


(104) 


The  crushing  strength  is,  of  course,  that  due  to  the  weaker 
metal,  whether  in  shaft  or  hub.     Let  Sc  be  this  least  safe  crushing 

LT 

fiber  stress.     The  crushing  strength  then  is  — ^-  Sc,  and,  equating 

external  force  to  internal  strength,  we  have: 

2PR      LT 


d 


or, 


rp    


4PR 


(105) 


The  proportions  of  the  key  must  be  such  that  the  equations  as 
above,  both  for  shearing  and  for  crushing,  shall  be  satisfied. 

PRACTICAL  MODIFICATION.  Pins  across  the  shaft  can  be 
used  to  drive  light  work  only,  for  the  shearing  area  cannot  be  very 
large.  A  large  pin  cuts  away  too  much  area  of  the  shaft,  decreas- 
ing the  latter's  strength,  Pins  are  useful  in  preventing  end  motion, 
but  in  this  case  are  expected  to  take  no  shear,  and  may  be  of  small 


164  MACHINE  DESIGN 

diameter.     The  common    split  pin  is  especially    adapted  to  this 
service,  and  is  a  standard  commercial  article. 

Taper  pins  are  usually  listed  according  to  the  Morse  standard 
taper,  proportions  of  which  may  be  found  in  any  handbook.  It 
is  desirable  to  use  standard  taper  pins  in  machine  construction,  as 
the  reamers  are  a  commercial  article  of  accepted  value,  and  readily 
obtainable  in  the  machine-tool  market. 

With  properly  fitted  keys,  the  shearing  strength  is  usually 
the  controlling  element.  For  shafts  of  ordinary  size,  the  standard 
proportions  as  given  in  tables  like  that  below  are  safe  enough 
without  calculation,  up  to  the  limit  of  torsional  strength  of  the 
shaft.  For  special  cases  of  short  hubs  or  heavy  loads,  a  calcula- 
tion is  needed  to  check  the  size,  and  perhaps  modify  it. 

Splines,   also   known    as    "  feather   keys,"  require  thickness 
greater  than  regular  keys,  on  account  of  the 
sliding  at  the  sides.     A  table  suggesting 
proportions  for  splines  is  given  on  page  166. 

Though  the  spline  may  be  either  in  the 
shaft  or  hub,  it  is  the  more  usual  thing  to 
find  the  spline  dovetailed  (Fig.  67«), 
"gibbed,"  or  otherwise  fastened  in  the 
hub;  and  a  long  spline  way  made  in  the 
shaft,  in  which  it  slides. 

The  straight  key,  accurately  fitted,  is  Fi£-  67a- 

the  most  desirable  fastening  device  for  ac- 
curate machines,  such  as  machine  tools,  on  account  of  the  fact  that 
there  is  absolutely  no  radial  force  exerted  to  throw  the  parts  out  of 
true.     It,  however,  requires  a  tight  fit  of  hub  to  shaft,  as  the  ksy 
cannot  be  relied  upon  to  take  up  any  looseness. 

The  taper  key  (Fig.  68),  by  its  wedging  action,  will  take  up 
some  looseness,  but  in  so  doing  throws  the  parts  out  slightly. 
Or,  even  if  the  bored  fit  be  good,  if  the  taper  key  be  not  driven 
home  with  care,  it  will  spring  the  hub,  and  make  the  parts  run 
untrue.  The  great  advantage,  however,  that  the  taper  key  has  of 
holding  the  hub  from  endwise  motion,  renders  it  a  very  useful 
and  practical  article.  It  is  usually  provided  with  a  head,  or  "  gib," 
which  permits  a  draw  hook  to  be  used  to  wedge  between  the  face 
of  the  hub  and  the  key  to  facilitate  starting  the  key  from  its  seatr 


MACHINE  DESIGN 


165 


Two  keys  at  90°  from  each  other  may  be  used  in  cases  where 
one  key  will  not  suffice.  The  fine  workmanship  involved  in 
spacing  these  keys  so  that  they  will  drive  equally  makes  this,  plan 
inadvisable  except  in  case  of  positive  and  unavoidable  necessity. 

The  "  Woodruff "  key  (Fig.  69)  is  a  useful  patented  article 
for  certain  locations.  This  key  is  a  half-disc,  sunk  in  the  shaft 


.       *                 v 

t 

i_ 

\~T|        H«       ^TAPER   i%      PER  FT. 

M     '                                   1 

Fig.  68. 

and  the  hub  is  slipped  over  it.  A  simple  rotary  cutter  is  dropped 
into  the  shaft  to  produce  the  key  seat;  and  on  account  of  the 
depth  in  the  shaft,  the  tendency  to  rock  sidewise  is  eliminated, 
and  the  drive  is  purely  by  shear. 

Keys  may  be  milled  out  of!  solid  stock,  or  drop-forged  to 
within  a  small  fraction  of  finished  size.  The  drop-forged  key  is 
an  excellent  modern  production  and  requires  but  a  minimum 


Fig.  69. 

amount  of  fitting.  Any  key,  no  matter  how  produced,  requires 
some  hand  fitting  and  draw  filing  to  bring  it  properly  to  its  seat 
and  give  it  full  bearing. 

It  is  good  mechanical  policy  to  avoid  keyed  fastenings 
whenever  possible.  This  does  not  mean  that  keys  may  never  be 
used,  but  that  a  key  is  not  an  ideal  way  to  produce  an  absolutely 
positive  drive,  partly  because  it  is  an  expensive  device,  and  partly 
because  the  tendency  of  any  key  is  to  work  itself  loose,  even  if 
carefully  fitted. 

The  following  tables  are  suggested  as  a  guide  to  proportions 


166 


MACHINE  DESIGN 


of  gib  keys  and  feather  keys,  and   will  be  found   useful   in  the 
absence  of  any  manufacturer's  standard  list: 

Fig.  70.   PROPORTIONS  FOR  GIB  KEYS. 


Diameter  of  shaft  (d),  inches. 

1 

1 

li 

if 

2 

2J 

31 

4 

5 

«* 

Width                     (W),  inches. 

156 

8 

H 

A 

J 

A 

1  1 

TB 

I 

iA 

lT5« 

If 

Thickness               (T),  inches. 

\ 

9 

Tl 

A 

if 

T76 

1  7 
~$T? 

11 

it 

1 

11 

Fig.  71.   PROPORTIONS  FOR  FEATHER  KEYS. 


.Diameter  of  Shaft    (d),  inches, 

I 

i 

11 

li 

2 

21 

2J 

3 

3} 

4: 

4J 

Width                     (W),  inches. 

A 

A 

i 

i 

5 
TTf 

1 

i 

* 

9 
TV 

rV 

1 

Thickness                (T),  inches. 

k 

T56 

I 

i 

T7. 

i 

i 

1 

3. 
4 

1 

i 

COTTERS. 

ANALYSIS.  Cotters  are  used  to  fasten  hubs  to  rods  rather 
than  shafts,  the  distinction  between  a  rod  and  a  shaft  being  that 
a  rod  takes  its  load  in  the  direction  of  its  length,  and  does  not 
drive  by  rotation.  A  cotter,  therefore,  is  nothing  but  a  cross-pin 
of  modified  form,  to  take  shearing  and  crushing  stress  in  the 
direction  of  the  axis  of  the  rod,  instead  of  perpendicular  to  it. 

Referring  to  Fig.  72,  one  will  see  that  the  cotter  is  made 
long  ana  thin— long,  m  order  to  get  sufficient  shearing  area  to  resist 
shearing  along  lines  A  and  B;  thin,  in  order  to  cut  as  little  cross- 
sectional  area  out  of  the  body  of  the  shaft  as  possible.  The  cotter 
itself  tends  to  shear  along  the  lines  A  and  B,  and  crush  along  the 
surfaces  K,  G,  and  J.  The  socket  tends  to  crush  along  the  surfaces 
K  and  G.  The  rod  end  D  tends  to  be  sheared  out  along  the  lines 
0  II  and  Q  E,  and  also  to  be  crushed  along  the  surface  J.  The 
socket  tends  to  be  sheared  alonor  the  lines  Y  U  and  X  Y. 

o 

The  cotter  is  made  taper  on  one  side,  thus  enabling  it  to  draw 
up  the  flange  of  the  rod  tightly  against  the  head  of  the  socket. 
This  taper  must  not  be  great  enough  to  permit  easy  "backing  out" 
and  loosening  of  the  cotter  under  load  or  vibration  in  the  rod.  In 
responsible  situations  this  cannot  be  safely  guarded  against  except 
through  some  auxiliary  locking  device,  such  as  lock  nuts  on  the 
end  of  the  cotter  (Fig.  73). 

THEORY.  Referring  to  Fig.  72,  assume  an  axial  load  of  Pr 
as  shown.  The  successive  equations  of  external  force  to  internal 


MACHINE  DESIGN 


167 


strength  are  enumerated  below,  for  the  different  actions  that  take 
place : 

For  shearing    along   lines  A  and  B,  w    being    the_average 
width  of  cotter,  and  Ss  safe  shearing  stress  of  cotter, 

(106) 


For  crushing  along  surfaces  K  and  G,  Sc  being   least   safe 
crushing  stress,  whether  of  cotter  or  socket, 


Pl=T(D1-D)Sc. 


(107) 


For  crushing  along   surface  J,  Sc   being  least  safe  crushing 
stress,  whether  of  cotter  or  socket, 


P,  ==  DTSC. 


(108) 


168 


MACHINE  DESIGN 


For  shearing  along  surfaces  CH  and  QE,  Ss  being  safe  ehea/ 
ing  stress  of  rod  end,  and  wl  end  of  slot  to  end  of  rod, 

P1  =  2«,1DSS.  (,09) 

For  tension  in  rod  end  at  section   across   slot,  St  being  safe 
tensile  stress  in  rod  end, 


(HO) 

For  tension  in  socket  at  section  across  slot,  St  being  safe  ten- 
sile  stress  in  socket, 


For  shearing  in  socket  along  the  lines  VU  and  XY,  S8  being 
safe  shearing  stress  in  the  socket,  and  w2  end  of  slot  to  end  of 
socket, 

P1  =  2w2(D1-D)Ss.  (112) 

The  proportions  of  cotter  and   socket   may  be   fixed   to  some 

extent  by  practical  or  as 
sumed  conditions.  The  di- 
mensions may  then  be  tested 
by  the  above  equations,  that 
the  safe  working  stresses  may 
not  be  exceeded,  the  dimen- 
sions being  then  modified  ac- 
cordingly. 

The  steel  of  which  both 
cotter  and  rod  would  ordina- 
rily be  made  has  range  of 
working  fiber  stress  as 
follows  : 

Tension,  3,000  to  12,000  (Ibs.  pel 

sq.  in.) 
Compression,  10,000  to  16,000  (Ibs. 

per  sq.  in.) 
Shear,  6,000  to  10,000   (Ibs.   per 

sq.  in.) 

The  socket,    if  made    of 


Fig.  73. 


cast  iron,  will  be  weak  as  regards  tension,  tendency  to  shear  out  at 


MACHINE  DESIGN 


169 


the  end,  and  tendency  to  split.  The  uncertainty  of  cast  iron  to 
resist  these  is  so  great  that  the  hub  or  socket  must  be  very  clumsy 
in  order  to  have  enough  surplus  strength.  This  is  always  a  notice- 
able feature  of  the  cotter  type  of  fastening,  and  cannot  well  be 
avoided. 

PRACTICAL  MODIFICATION.  The  driving  faces  of  the  cot- 
ter are  often  made  semicircular.  This  not  only  gives  more  shear- 
ing area  at  the  sides  of  the  slots,  but  makes  the  production  of  the 
slots  easier  in  the  shop.  It  also  avoids  the  general  objection  to 
sharp  corners — namely,  a  tendency  to  start  cracks. 

A  practicable  taper  for 
cotters  is  \  inch  per  foot. 
This  will  under  ordinary 
circumstances  prevent  the 
cotter  from  backing  out 
under  the  action  of  the  load. 
When  set  screws  against 
the  side  of  the  cotter,  or 


lock  nuts  are  used,  as  in 
Fig.  73,  the  taper  may  be 
greater  than  this,  perhaps 
as  much  as  1J  inches  per 
foot. 

In  the  common  use  of 
the  cotter  for  holding  the  Fig.  74. 

strap  at  the  ends  of  con- 
necting rods,  the  strap  acts  like  a  modified  form  of  socket.  This  is 
shown  in  Figs.  73  and  74.  Here,  in  addition  to  holding  the  strap 
and  rod  together  lengthwise,  it  may  be  necessary  to  prevent  their 
spreading,  and  for  this  purpose  an  auxiliary  piece  G  with  gib  ends 
is  used.  The  tendency  without  this  extra  piece  is  shown  by  the 
dotted  lines  in  Fig.  74. 

The  general  mechanical  fault  with  cottered  joints  is  that  the 
action  of  the  load,  especially  when  it  constantly  reverses,  as  in 
pump  piston  rods,  always  tends  to  work  the  cotter  loose.  Yibra- 
tion  also  tends  to  produce  the  same  effect.  Once  this  looseness  is 
started  in  the  joint,  the  cotter  loses  its  pure  crushing  and  shearing 
action,  and  begins  to  partake  of  the  nature  of  a  hammer,  and 


170  MACHINE  DESIGN 

pounds  itself  and  its  bearing  surfaces  out  of  their  true  shape. 
Instead  of  a  collar  on  the  rod,  we  often  find  a  taper  fit  of  the  rod 
in  the  socket;  and  any  looaeness  in  this  case  is  still  worse,  for  the 
rod  then  has  end  play  in  the  socket,  and  by  its  "  shucking  "  back 
and  forth  tends  to  split  open  the  socket. 

The  only  answer  to  these  objections  is  to  provide  a  positive 
locking  device,  and  take  up  any  looseness  the  instant  it  appears. 

PROBLEMS  ON  KEYS,  PINS,  AND  COTTERS. 

1.  Calculate  the  safe  load  in  shear  which  can  be  carried  on  a  key 

4  inch  wide,  f  inch  thick,  and  5  inches  long.     Assume  Ss  =  6,000. 

2.  Assuming  the  above  key  to  be  T3^  inch  in  hub  and  T?6  inch 
in  shaft,  test  its  proportions  for  crushing,  at  Sc  =  16,000. 

3.  A  gear  60  inches  in  diameter  has  a  load  of  3,000  Ibs.  at 
the   pitch   line.     The    shaft  is    4  inches  in  diameter,  in  a  hub, 

5  inches  long;  and  the  key  is  a  standard  gib  key  as  given  in  the 
table.     Test  its  proportions  for  shearing. 

4.  A  piston  rod  2  inches  in  diameter  carries  a  cotter  §  inch 
thick,  and  has  an  axial  load  of  20,000  Ibs.      Calculate  the  average 
width  of  the  cotter.     Ss  =  9,000. 

5.  Calculate  fiber  stress    in   rod  in    preceding  problem    at 
section  through  slot. 

6.  How  far  from  the  end  of  rod  must  the  end  of  slot  be  ? 

7.  Calculate  the  crushing  fiber  stresses  ,on  cotter,  rod,  and 
socket. 

8.  How  far  from  the  end  of  socket  must  the  end  of  slot  be, 
assuming  the  socket  to  be  of  steel  ? 

BEARINGS,  BRACKETS,  AND  STANDS. 

NOTATION— The  following  notation  is  used  throxighout  the  chapters  on  Bearings 
Brackets,  and  Stands. 

A  =  Area  (square  inches).  K  —Number  of  reroiutions  per  minute. 

a  =  Distance  between  bolt  centei-s  n  —  Number  of  bolts  in  cap. 

(inches).  wi  =  Number  of  bolts  in  bracket  base, 

ft  =  Width  of  bracket  base  (inches).  P  =  Total  pressure  on  bearing  (Ibs.). 

c  =  Distance  of  neutral  axis  from  outer  p  =  Pressure  per  square   inch  of  pro- 
fiber  (inches).  jected  area  (Ibs). 

D  =  Diameter  of  shaft  (inches).  S  =  Safe  tensile  fiber  stress  (Ibs.). 

d  =  Diameter  of  bolt  body  (inches).  Ss  =    "     shearing      "  (Ibs.). 

di  =  Diameter  at  root  of  thread  (inches) .  T  =  'Total  load  on  bolts  at  top  of 
H  =  Horse-power.  bracket  (Ibs.), 

h  =  Thickness  of  cap  at  center  (inches).  t    =  Thickness  of  bracket  base  (inches). 

I  =  Moment  of  inertia.  x  =  Distance  from  line  of  action  of  load 
L  =  Length  of  bearing  (inches).  to  any  section  of  bracket  (inches). 

t=  Coefficient  of  friction  (per  cent). 


MACHINE  DESIGN  171 

ANALYSIS.  Machine  surfaces  taking  weight  and  pressure 
of  other  parts  in  motion  upon  them  are,  in  general,  known  as 
bearings.  If  the  motion  is  rectilinear,  the  bearing  is  termed  a 
slide,  guide,  or  way,  such' as  the  cross  slide  of  a  lathe,  the  cross- 
head  guide  of  a  steam  engine,  or  the  ways  of  a  lathe  bed. 

If  the  motion  is  a  rotary  one,  like  that  of  the  spindle  of  a  lathe, 
the  simple  word  "  bearing  "  is  generally  used. 

In  any  bearing,  sliding  or  rotary,  there  must  be  strength  to 
carry  the  load,  stiffness  to  distribute  the  pressure  evenly  over  the 
full  bearing  surface,  low  intensity  of  such  pressure  to  prevent  the 
lubricant  from  being  squeezed  out  and  to  minimize  the  wear,  and 
sufficient  radiating  surface  to  carry  away  the  heat  generated  by 
friction  of  the  surfaces  as  fast  as  it  is  generated.  Sliding  bearings 
are  of  such  varied  nature,  and  exist  under  conditions  so  peculiar 
to  each  case,  that  a  general  analysis  is  practically  impossible 
beyond  that  given  in  the  sentence  above. 

Rotary  bearings  can  be  more  definitely  studied,  as  there  are 
but  two  variable  dimensions,  diameter  and  length,  and  it  is  the 

O        ' 

proper  relation  between  these  two  that  determines  a  good  bearing. 
The  size  of  the  shaft,  as  noted  under  "  Shafts,"  is  calculated  by 
taking  the  bending  moment  at  the  center  of  the  bearing,  combin- 
ing it  with  the  twisting  moment,  and  solving  for  the  diameter 
consistent  with  the  assumed  fiber  stress.  But  this  size  must  then 
be  tried  for  deflection  due  to  the  bending  load,  in  order  that  the 
requirement  for  stiffness  may  be  fulfilled.  When  this  is  accom- 
plished, the  friction  at  the  bearing  surface  may  still  generate  so 
much  heat  that  f;he  exposed  surface  of  the  bearing  will  not  radiate 
it  as  fast  as  generated,  in  which  case  the  bearing  gets  hotter  and 
hotter,  until  it  finally  burns  out  the  lubricant  and  melts  the  lining 
of  the  bearing,  and  ruin  results.  * 

The  heat  condition  is  usually  the  critical  one,  as  it  is  very 
easy  to  make  a  short  bearing  which  is  strong  enough  and  amply 
stiff  for  the  load  it  carries,  but  whjch  nevertheless  is  a  failure  as 
a  bearing,  because  it  has  so  small  a  radiating  surface  that  it  can- 
not run  cool. 

The  side  load  which  causes  the  friction  and  the  consequent 
development  of  heat,  is  due  to  the  pull  of  the  belt  in  the  case  of 
pulleys,  the  load  on  the  teeth  of  gears,  the  puii  on  cranks  and 


172  MACHINE  DESIGN 


levers,  the  weight  of  parts,  etc.  If  we  could  exert  pure  torsion  on 
shafts  without  any  side  pressure,  and  counteract  all  the  weight 
that  comes  on  the  shaft,  we  should  not  have  any  trouble  with  the 
development  of  heat  in  bearings;  in  fact,  there  would  theoretically 
be  no  need  of  bearings,  as  the  shafts  would  naturally  spin  about 
their  axes,  and  would  not  need  support. 

It  can  be  shown,  theoretically,  that  the  radiating  surface  of  a 
bearing  increases  relatively  to  the  heat  generated  by  a  given  side 
load,  only  when  the  length  of  the  hearing  is  increased.  In  other 
words,  increasing  the  diameter  and  not  the  length,  theoretically 
increases  the  heat  generated  per  unit  of  time  just  as  much  as  it 
increases  the  radiating  surface;  hence  nothing  is  gained,  and  heat 
accumulates  in  the  bearing  as  before.  This  important  fact  is  veri- 
fied by  the  design  of  high-speed  bearings,  which,  it  is  always 
noted,  are  very  long  in  proportion  to  their  diameter,  thus  giving 
relatively  high  radiating  power. 

Bearings  must  be  rigidly  fastened  to  the  body  of  the  machine 
in  some  way,  and  the  immediate  support  is  termed  a  bracket, 
frame,  or  housing.  "Bracket"  is  a  very  general  term,  and  ap- 
plies to  the  supports  of  other  machine  parts  besides  "bearings/9 
It  is  especially  applicable  to  the  more  familiar  types  of  bearing 
supports,*  and  is  here  introduced  to  make  the  analysis  complete. 

The  bracket  must  be  strong  enough  as  a  beam  to  take  the 
side  load,  the  bending  moment  being  figured  at  such  points  as  are 
necessary  to  determine  its  outline.  It  may  be  of  solid,  box,  or 
ribbed  form,  the  latter  being  the.  most  economical  of  material,  and 
usually  permitting  the  simplest  pattern.  The  fastening  of  the 
bracket  to  the  main  body  of  the  machine  must  be  broad  to  give 
stability;  the  bolts  act  partly  in  shear  to  keep  the  bracket  from 
sliding  along  its  base,  and  partly  in  tension  to  resist  its  tendency 
to  rotate  about  some  one  of  its  edges,  due  to  the  side  pull  of  the 
belt,  gear  tooth,  or  lever  load,  as  the  case  may  be.  The  weight  of 
the  bracket  itself  and  of  the  parts  it  sustains  through  the  bearing, 
has  likewise  to  be  considered;  and  this  acts,  in  conjunction  with 
the  working  load  on  the  bearing,  to  modify  the  direction  and 
magnitude  of  the  resultant  load  on  the  bracket  and  its  fastening. 

Stands  are  forms  of  brackets,  and  are  subject  to  the  same 
analysis.  The  distinction  is  by  no  means  well  defined,  although 


MACHINE  DESIGN 


173 


we  usually  think  more  readily  of  a  stand  as  having  an  upright  or 
inverted  position  with  reference  to  the  ground.  The  ordinary 
'"  hanger  "  is  a  good  example  of  an  inverted  stand;  and  the  regular 
"  floor  stand,"  found  on  jack  shafts  in  some  power  houses,  is  an 
example  of  the  general  class. 

THEORY.  As  the  method  of  calculation  of  the  diameter  of 
the  shaft,  as  well  as  its  deflection,  has  been  considered  under 
"  Shafts,"  we  may  assume  that  the  theoretical  study  of  bearings 
starts  on  a  given  basis  of  shaft  diameter  D.  The  main  problem 
then  being  one  of  heat  control,  let  us  first  calculate  the  amount  of 
heat  developed  in  a  bearing  by  a  given  side  load.  The  force  of 
friction  acts  at  the  circumference  of  the  shaft,  and  is  equal  to  the 
coefficient  of  friction  times  the  normal  force;  or,  for  a  given  side  load 
P,  Fig.  75,  the  force  of  friction 
would  be  jjiP.  The  peripheral 
speed  of  the  shaft  for  N  revolu- 

7TDN 

tions  per  minute  is  —  y^-     feet 

per  minute.     As  work  is  "force 
times  distance,"  the  work  wasted 


in  friction  is  then 


foot- 


pounds  per  minute.  One  horse- 
power being  equal  to  33,000  foot- 
pounds per  minute,  we  have  the 
equation, 

H  = 


Fig.  75. 


12  X  33,000 

The  value  of  jut  for  ordinary,  well-lubricated  bearings,  may  run  as 
low  as  5  per  cent;  but  as  the  lubrication  is  often  impaired,  it 
quite  commonly  rises  to  10  or  12  per  cent.  A  value  of  8  per  cent 
is  a  fair  average.  This  amount  of  horse-power  is  dissipated 
through  the  bearing  in  the  form  of  heat.  If  we  could  exactly 
determine  the  ability  that  each  particle  of  the  metal  around  the 
shaft  had  to  transmit  the  heat,  or  to  pass  it  along  to  the  outside 
of  the  casting,  and  if  we  could  then  determine  the  ability  of  the 
particles  of  air  surrounding  the  casting  to  receive  and  carry  away 


174  MACHINE  DESIGN 

this  heat,  we  could  calculate  just  such  proportions  of  the  bearing 
and  its  casing  as  would  never  choke  or  retard  this  free  transfer  of 
heat  away  from  the  running  surface. 

Such  refined  theory  is  not  practical,  owing  to  the  complicated 
shapes  and  conditions  surrounding  the  bearing.  The  best  that  we 
can  do  is  to  say  that  for  the  usual  proportions  of  bearings  the  side 
load  may  exist  up  to  a  certain  intensity  of  "  pressure  per  square 
inch  of  projected  area  "  of  bearing,  or,  in  form  of  an  equation, 


(,14) 

The  constant  p  is  of  a  variable  nature,  depending  on  lubrication, 
speed,  air  contact,  and  other  special  conditions.  For  ordinary 
bearings  having  continuous  pressure  in  one  direction,  and  only 
fair  lubrication,  400  to  500  is  an  average  value.  When  the  pres- 
sure changes  direction  at  every  half-revolution,  the  lubricant  has 
a  better  chance  to  work  fully  over  the  bearing  surface,  and  a 
higher  value  is  permissible,  say,  500  to  800.  In  locations  where 
mere  oscillation  takes  place,  not  continuous  rotation,  and  reversal 
of  pressure  occurs,  as  on  the  cross-head  pin  of  a  steam  engine,  p 
may  run  as  high  as  900  to  1,200.  On  the  crank  pins  of  locomo- 
tives, which  have  the  reversal  of  pressure,  and  the  benefit  of  high 
velocity  through  the  air  to  facilitate  cooling,  the  pressures  may  run 
equally  high.  On  the  eccentric  crank  pins  of  punching  and  shear- 
ing machines,  wrhere  the  pressure  acts  only  for  a  brief  instant  and 
at  intervals,  the  pressure  ranges  still  higher  without  any  dangerous 
heating  action. 

When  a  bearing,  for  practical  reasons,  is  provided  with  a  cap 
held  in  place  by  bolts  or  studs,  the  theory  of  the  cap  and  bolts  is 
of  little  importance,  unless  the  load  comes  directly  against  the  cap 
and  bolts.  Except  in  the  latter  case,  the  proportions  of  the  cap  and 
the  size  of  the  bolts  are  dependent  upon  general  appearance 
and  utility,  it  being  manifestly  desirable  to  provide  a  substantial 
design,  even  though  some  excess  of  strength  is  thereby  introduced. 

For  the  worst  case  of  loading,  however,   which  is  when  the 

•  o' 

cap  is  acted  upon  by  the  direct  load,  such  as  P  in  Fig.  76,  we  have 
the  condition  of  a  centrally  loaded  beam  supported  at  the  bolts. 
It  is  probable  that  the  beam  is  partially  fixed  at  the  ends  by  the 
clamping  of  the  nut;  also  that  the  load  P,  instead  of  being  con- 


MACHINE  DESIGN 


175 


centrated  at  the  center,  is  to  some  extent  distributed.    It  is  hardly 

Pa          Pa  A, 

fair  to  assume  the  external   moment   equal   to  -^~  or  -j^» tne  one 

being  too  small,  perhaps,  and  the  other  too  large.     It  will  be   rea- 
sonable to  take  the  external  moment  at  —7-,  in  which  case,  equat- 


Y 

r«* 

fri 

m 

t: 

r-_-j 

1 

* 

E 

Fig.  76. 

ing  the  external  moment  to  the  internal  moment  of  resistance, 

Pa  _  SI  __  SLA2 

O  0 •  O 

from  which,  the  length  of  bearing  being  known,  we  may  calculate 
the  thickness  A. 

One  bolt  on  each  side  is  sufficient  for  bearings  not  more 
than  G  inches  long,  but  for  longer  bearings  we  usually  find  two 
bolts  on  a  side.  The  theoretical  location  for  two  bolts  on  a  side, 
in  order  that  the  bearing  may  be  equally  strong  at  the  bolts  and 
at  the  center  of  the  length,  may  be  shown  by  the  principles  of 

mechanics    to  be  -^  L  from   each  end,  as  indicated  in  Fig.  76. 
The  bolts  are  evidently  in  direct  tension,  and  if  equally  loaded 


176  MACHINE  DESIGN 

would  each  take  their  fractional  share  of  the  whole  load  P.     This 

2 
is  difficult  to  guarantee,  and  it  is  safer  to  consider  that  -^  P  may 

be  taken  by  the  bolts  on  one  side.  On  this  basis,  for  total  number 
of  bolts  n,  equating  the  external  force  to  the  internal  resistance  of 
the  bolts,  we  have  : 

2  STrdS       n 


from  which  the  proper  commercial  diameter  may  be  readily  found. 
The  bracket  may  have  the  shape  shown  in  Fig.  77.  The 
portion  at  B  is  under  direct  shearing  stress;  and  if  A  be  the  area 
at  this  point,  and  Sfl  the  safe  shearing  stress,  then,  equating  the 
external  force  to  the  internal  shearing  resistance, 

P=ASS.  (117) 

The  same  shear  comes  on  all  parts  of  the  bracket  to  the  left  of  the 
load,  but  there  is  an  excess  of  shearing  strength  at  these  points. 

At  the  point  of  fastening,  the  bolts  are  in  shear,  due  to  the 
same  load,  for  which  the  equation  is 

A-  (n8) 

For  the  upper  bolts,  the  case  is  that  of  direct  tension,  assum- 
ing that  the  whole  bracket  tends  to  rotate  about  the  lower  edge  E. 
To  find  the  load  T  on  these  bolts,  we  should  take  moments  about 
the  point  E,  as  follows: 

PT 

(119) 


Then,  equating  the  external  force  to  the  internal  resistance, 

PLt       7T^2      n. 

—       -J-X-jffl.  (120) 

The  upper  flange  is  loaded  with  the  bolt  load  T,  and  tends  to 
break  off  at  the  point  of  connection  to  the  main  body  of  the 
bracket,  the  external  moment,  therefore,  being  Tr.  The  section 
of  the  flange  is  rectangular;  hence  the  equation  of  external  and  in- 
ternal  moments  is: 


MACHINE  DESIGN 


177 


PI* 


(121) 


It  ma^  be  noted  that  the  lower  bolts  act  on  such  a  small  leverage 
about  E,  that  they  would  stretch  and  thus  permit  all  the  load  to 
be  thrown  on  the  upper  bolts;  this  is  the  reason  why  they  are  not 
subject  to  calculation  for  tension. 


ip 


SCI 


Fig.  77. 

The  section  of  the  bracket  to  the  left  of  the  load  P  is  depend- 
ent upon  the  bending  moment,  for,  if  this  section  is  large  enough 
to  take  the  bending  moment  properly,  the  shear  may  be  disregard- 
ed. It  should  be  calculated  at  several  points,  to  make  sure  that 
the  fiber  stress  is  within,  allowable  limits.  The  general  expression 
for  the  equation  of  moments  is,  for  any  section  at  leverage  #, 

P»=— ,  (122) 


from  which,   by   the    proper  substitution  of  the  moment  of  in- 


178 


MACHINE  DESIGN 


ertia  of  the  section,  the  fiber  stress  can  be  calculated.  The  mo- 
ment of  inertia  for  simple  ribbed  sections  can  be  found  in  most 
handbooks.  The  process  of  solution  of  the  above  equation,  though 

simple,  is  apt  to  be  tedious, 
and  'is  not  considered  neces- 
sary to  illustrate  here. 

PRACTICAL  MODIFN 
CATION.  Adjustment  is  an 
important  practical  feature  of 
bearings.  Unless  the  propor- 
tions are  so  ample  that  wear 
is  inappreciable,  simple  and 
ready  adjustment  must  be 
provided.  The  taper  bush- 
ing. Fig.  79,  is  neat  and  sat- 
isfactory for  machinery  in 
which  expense  and  refinement 

F-    •„  are  permissible.    This  is  true 

of  some  machine  tools,  but  is 

not  true  of  the  general  "  run  "  of  bearings.  The  most  common 
form  of  adjustment  is  secured  by  the  plain  cap  (which  may  or  may 


Fig.  79. 


not  be  tongued  into  the  bracket).,  with  liners  placed  in  the  joint 
when  new,  which  may  subsequently  be  removed  or  reduced  so  as 
to  allow  the  cap  to  close  down  upon  the  shaft.  Several  forms  of 
cap  bearings  are  illustrated  in  Figs.  80,  81,  and  82. 


MACHINE  DESIGN 


179 


Large  engine  shaft  bearings  have  special  forms  of  adjustment 
by  means  of  wedges  and  screws,  which  take  up  the  wear  in  all 
directions,  at  the  same  time  accurately  preserving  the~aKgnment 
of  the  shafts;  but  this  refinement  is  seldom  required  for  shafts  of 
ordinary  machinery. 

In  cases  where  the  cap  bearing  is  not  applicable,  a  simple 
bushing  may  be  used.  This  may  be  removed  when  worn,  and  a 


as  the 
regard  - 

in  that 


Fig.  80.  Fig.  81. 

new  one  inserted,  the  exact  alignment  being  maintained, 
outside  will  be  concentric  with  the  original  axis  of  shaft, 
less  of  the  wear  which  has  taken  place  in  the  bore. 

The  lubrication  of  bearings  is  a  part  of  the  design, 
the  lubricant  should  be  intro- 
duced at  the  proper  point,  and 
pains  taken  to  guarantee  its  dis- 
tribution to  all  points  of  the  run- 
ning surface.  The  method  of 
lubrication  should  be  so  certain 
that  no  excuse  for  its  failure 
would  be  possible.  Grease  is  a 
successful  lubricator  for  heavy 
loads  and  slow  speeds,  oil  for 
light  loads  and  high  speeds. 

In  order  to  insure  the  lubri-  pjg  82. 

cant  reaching   the   sliding    sur- 
faces and  entering  between  them,  it  must  be  introduced  at  a  point 


180  MACHINE  DESIGN 

where  the  pressure  is  moderate,  and  where  the  motion  of  the  parts 
will  naturally  lead  it  to  all  points  of  the  bearing.  Grooves  and 
channels  of  ample  size  assist  in  this  regard.  A  special  form  of 
bearing  uses  a  ring  riding  on  the  shaft  to  carry  the-  oil  constantly 
from  a  small  reservoir  beneath  the  shaft  up  to  the  top,  where  it  is 
distributed  along  the  bearing  and  finally  flows  back  to  the  reser- 
voir and  is  used  again. 

The  materials  of  which  bearings  are  made  vary  with  the 
service  required  and  with  the  refinement  of  the  bearing.  Cast  iron 
makes  an  excellent  bearing  for  light  loads  and  slow  speeds,  but 
it  is  very  apt  to  "  seize  "  the  shaft  in  case  the  lubrication  is  in  the 
least  degree  impaired.  Bronze,  in  its  many  forms  of  density  and 
hardness,  is  extensively  used  for  high-grade  bearings,  but  it  also 
has  little  natural  lubricating  power,  and  requires  careful  attention 
to  keep  it  in  good  condition. 

Babbitt,  a  composition  metal,  of  varying  degrees  of  hardness, 
is  the  most  universal  and  satisfactory  material  for  ordinary  bear- 
ings. It  affords  a  cheap  method  of  production,  being  poured  in 
molten  form  around  a  mandrel,  and  firmly  retained  in  its  casing  or 
shell  through  dovetailed  pockets  into  which  the  metal  flows  and 
hardens.  It  requires  no  boring  or  extensive  fitting.  Some 
scraping  to  uniform  bearing  is  necessary  in  most  cases,  but  this  is 
easily  and  cheaply  done.  Babbitt  is  a  durable  material,  and  has 
some  natural  lubricating  power,  so  that  it  has  less  tendency  to 
heat  with  scanty  lubrication  than  any  of  the  materials  previously 
mentioned.  Almost  any  grade  of  bearing  may  be  produced  with 
babbitt.  In  its  finest  form  the  babbitt  is  hammered,  or.pened, 
into  the  shell  of  the  bearing,  and  then  bored  out  nearly  to  size,  a 
slightly  tapered  mandrel  being  subsequently  drawn  through,  com- 
pressing  the  babbitt  and  giving  a  polished  surface. 

A  combination  bearing  of  babbitt  and  bronze  is  sometimes 
used.  In  this  the  bronze  lies  in  strips  from  end  to  end  of  the 
bearing,  and  the  babbitt  fills  in  between  the  strips.  The  shell, 
oeing  of  bronze,  gives  the  required  stiffness,  and  the  babbitt  the 
favorable  running  quality. 

PROBLEMS  ON  BEARINGS,    BRACKETS,  AND  STANDS. 

1.     The  allowable  pressure  on  a  bearing  is  300  pounds  per 


MACHINE  DESIGN  181 

square  inch  of  projected  area.  What  is  the  required  length  of 
the  bearing  if  the  total  load  is  4,500  pounds  and  the  diameter  is 
3  inches  ? 

2.  The  cross -head  pin  of  a  steam  engine  must  be  2.5  inches 
in  diameter  to  withstand  the  shearing  strain.     If  the  maximum 
pressure  is  10,000  pounds,  what  length  should  be  given  to  the  pin  ? 

3.  The  journals  on  the  tender  of  a  locomotive  are  3J  X  7 
inches.     The  total  weight  of  the  tender  and  load  is  60,000  pounds. 
If  there  are  8  journals,  what  is  the  pressure  per  square  inch  of 
projected  area  ? 

4.  What  horse-power  is  lost  in  friction  at  the  circumference 
of  a  3-inch  bearing  carrying  a  load  of  6,000  pounds,  if  the  number 
of  revolutions  per  minute  is  150  and  the  coefficient  of  friction  is 
assumed  to  be  5  per  cent  ? 

5.  "The  cast-iron  bracket  in  Fig.   77  has  a  load  P  of  1,000 
pounds.     Determine  the  fiber  stress  in  the  web  section  at  the  base 
of  the    bracket  if  the  thickness  is  taken  at  ^  inch,  and  Lx  =  12 
inches;  I  =  20  inches;  Jc  =  11  inches;  t  =  1  inch. 

6.  Calculate    the   diameter   of  the  bolts  at  the  top  of  the 
bracket. 

7.  Assuming  /-equal  to  6  inches,  what  is  the  fiber  stress  at 
the  root  of  flange  ? 


INDEX 


Page 

Application  of  to  practical  case 23 

Base 11 

Beams 16 

Bearings 170 

length  of 32 

problems  on 180 

Belting 

horse-power  transmitted  by 81 

material  of 83 

speed  of 83 

Belts 

analysis  of 75 

initial  tension  in 84 

problems  on '. .  .- . 85 

theory  of 77 

width  of 31 

Bending  combined  with  torsion 106 

Bessemer  steel 113 

Bevel  gears 1 26 

Bolts -. 149 

problems  on 159 

Brackets 172 

problems  on 180 

Brackets  and  caps 47 

Brake-relief  spring 57 

Brake-strap  bracket . . ' 57 

Calculations,  notes,  and  records 8 

Cap  screws 158 

Centrifugal  whirling 109 

Clamp  coupling 148 

Classification  of  machinery 62 

Claw  coupling 148 

Combined  stresses 103 

Compression 17 

combined  with  torsion 105 

Conditions  and  forces,  analysis  of 10 

Constructive  mechanics 16 

Cotters • 166 

problems  on 170 

Couplings 145 

problems  on 149 


184  INDEX 


Page 

Cycloidal  curves 116 

Data  on  sketch 32 

Definition  of  machine  design 3 

Deflection 107 

Delineation 14 

Design,  method  of • 10 

Driving  gears 26 

Drum  and  brake '. 51 

Drum  shaft ,  .  .  f 37 

Empirical  data 7 

Flange  coupling 146 

Foot  lever 57 

Forces .  16 

Friction 19 

Friction  clutches 139 

problems  on 145 

Gear  guard 57 

Gears 43 

problems  on .  137 

General  drawing 58 

Handbooks 7 

Height  of  centers '. 32 

Hook-tooth  gear •. 124 

Horse-power  of  shafting 110 

Horse-power  transmitted  by  belting 81 

Initial  tension  in  belt 84 

Invention • 7 

Involute  curves 116 

Keys  and  pins ....  160 

problems  on 170 

Leather  belting,  strength  of 81 

Length  of  bearings 32 

Lock  nuts •* 157 

Lubrication 19 

Machine  screws • 159 

Machine  tools 63 

Machinery 

classification  of 62 

mill  and  plant 69 

motive  power 65 

structural 66 

Material  of  belting ....  83 

Mechanical  development 3 

Mechanical  specification 

Mechanical  thought 3 

Method  of  design 10 

Mill  and  plant  machinery 69 

Moments •  •  •  • 16 

Mortise  teeth 123 


INDEX  185 


Page 

Motive-power  machinery ....." 65 

Muff  coupling 147 

Nickel  steel 113 

Object  of  machine  design 77  ..7T".TTn--.       3 

Open-hearth  steel 113 

Original  design,  pointed  suggestions  on.  . 70 

Pinion  bore 37 

Pinion  shaft  outer  bearing 37 

Pitch  cylinders .  . 116 

Practical  modification 13 

Preliminary  layout 39 

Preliminary  sketch 26 

Pressure  combined  with  bending 104 

Problem  in  machine  design '. 25 

Problems  on 

bearings,  brackets,  and  stands 180 

belts 85 

bolts,  studs,  nuts,  and  screws 159 

couplings 149 

friction  clutches 145 

gears 137 

keys,  pins,  and  cotters 170 

shafts . 114 

Pulley  arms 89 

Pulley  hub , 91 

Pulley  rim ; 87 

Pulleys 29,41,86 

problems  on 99 

special  forms  of 98 

Rope  and  drum 26 

Set  screws 158 

Shaft  outside  of  pinion 37 

Shafts 100 

problems  on • 114 

sizes  of 32 

Shrouding  a  tooth .  .  .  ; 123 

Simple  bending 103 

Simple  torsion 103 

Sizes  of  shafts 32 

Specification 14 

Speed  of  belting 83 

Spline 161 

Split  pulleys 94 

Spur  gear  arms 122 

Spur  gear  hub 1 23 

Spur  gear  rim „ 121 

Spur  gears 114 

Stands 172 

problems  on 180 


186  INDEX 


Page 

Strain -...-..• 21 

Strength  of  leather  belting '. 81 

Stress 21 

Structural  machinery . 66 

Stub  tooth 124 

Studs 149,  158 

problems  on :  . 1 59 

Tables 

bolts,  strength  of 154 

feather  keys,  proportions  for 166 

gear  design  data - 1 26 

gib  keys,  proportions  for 166 

leather  belting,  sizes  of 84 

torsional  moments 31 

Tabulation  of  torsional  moments : 31 

Tap  bolts 158 

Tension 17 

combined  with  bending •. 104 

combined  with  torsion .  105 

Theoretical  design 12 

Through  bolts : 158 

Torsion ' 17 

Torsional  moments,  tabulation  of 31 

Web  gears 125 

Width  of  belt 31 

Working  stresses . 20 

Worm  ard  worm  gear 132 


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TOOL  MAKING.  By  E.R.Markham.  200  pp., 325  illus. 
How  to  make,  how  to  use  tools.  Profusely 
illustrated.  Price $1.50 

MACHINE  DESIGN.  By  C.  L.  Griffin.  200  PP., 
82  designs.  Written  by  one  of  the  foremost 
authorities  of  the  day.  Every  illustration 
represents  a  new  device  in  machine  shop 
practice.  Price $  1.50 


These  volumes  are  handsomely  bound  in  red  art  Vellum  de  Luxe,  size  6K  x  9%  inches.  Sent 
prepaid  to  any  part  of  the  world,  on  receipt  of  price.  Remit  by  Draft,  Postal  Order,  Express  Order, 
or  Registered  Letter. 


AMERICAN  SCHOOL  OF  CORRESPONDENCE,  CHICAGO 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 


DAY    AND    TO    $1.OO    ON    THE    SEVENTH     DAY 
OVERDUE. 

SEP  281937 

Ort   23tt*» 

MAY    1    W 

OCf  251948 

• 

LD  21-95m-7,'37 

